Are we exaggerating importance of model assumption and evaluation in an era when analyses are often carried out by laymen Bottom line, the more I learn about statistics, the less I trust published papers in my field; I simply believe that researchers are not doing their statistics well enough.

I'm a layman, so to speak. I'm trained in biology but I have no formal education in statistics or mathematics. I enjoy R and often make an effort to read (and understand...) some of the theoretical foundations of the methods that I apply when doing research. It wouldn't surprise me if the majority of people doing analyses today are actually not formally trained. I've published around 20 original papers, some of which have been accepted by recognized journals and statisticians have frequently been involved in the review-process. My analyses commonly include survival analysis, linear regression, logistic regression, mixed models. Never ever has a reviewer asked about model assumptions, fit or evaluation.
Thus, I never really bothered too much about model assumptions, fit and evaluation. I start with a hypothesis, execute the regression and then present the results. In some instances I made an effort to evaluate these things, but I always ended up with "well it didn't fulfill all assumptions, but I trust the results ("subject matter knowledge") and they are plausible, so it's fine" and when consulting a statistician they always seemed to agree.
Now, I've spoken to other statisticians and non-statisticians (chemists, physicians and biologists) who perform analyses themselves; it seems that people don't really bother too much about all these assumptions and formal evaluations. But here on CV, there is an abundance of people asking about residuals, model fit, ways to evaluate it, eigenvalues, vectors and the list goes on. Let me put it this way, when lme4 warns about large eigenvalues, I really doubt that many of its users care to address that...
Is it worth the extra effort? Is it not likely that the majority of all published results do not respect these assumptions and perhaps have not even assessed them? This is probably a growing issue since databases grow larger every day and there is a notion that the bigger the data, the less important is the assumptions and evaluations.
I could be absolutely wrong, but this is how I have perceived this.
Update:
Citation borrowed from StasK (below): http://www.nature.com/news/science-joins-push-to-screen-statistics-in-papers-1.15509
 A: The nature of violations of assumptions can be an important clue for future research. For example, a violation of the proportional-hazards assumption in Cox survival analysis might be due to a variable with a large effect on short-term survival but little effect in the longer term. That's the type of unexpected but potentially important information you can get by examining the validity of your assumptions in a statistical test.
So you do yourself, not just the literature, a potential disservice if you don't test the underlying assumptions. As high-quality journals start requiring more sophisticated statistical review you will find yourself called on more frequently to do so. You don't want to be in a position where a test required by a statistical reviewer undermines what you thought had been a key point of your paper.
A: I'll answer from an intermediate perspective. I'm not a statistician, I'm chemist. However, I've spent the last 10 years specializing in chemometrics = statistical data analysis for chemistry-related data. 

I simply believe that researchers are not doing their statistics well enough. 

That is probably the case.

Short version: 
Now about the assumptions. IMHO the situation here is far too heterogeneous to deal with it in one statement. Understanding of both what exactly the assumption is needed for and in which way it is likely to be violated by the application is necessary in order to judge whether the violation is harmless or critical. And this needs both the statistics as well as the application knowledge.
As a practitioner facing unachievable assumptions, however, I need something else as well: I'd like to have a "2nd line of defense" that e.g. allows me to judge whether the violation is actually causing trouble or whether it is harmless. 

Long version:


*

*From a practical point of view, some typical assumptions are almost never met. Sometimes I can formulate sensible assumptions about the data, but often then the problems become so complicated from a statistical point of view that solutions are not yet known. By now I believe that doing science means that you'll hit the borders of what is known likely not only in your particular discipline but maybe also in other disciplines (here: applied statistics).

*There are other situations where certain violations are known to be usually harmless - e.g. the multivariate normality with equal covariance for LDA is needed to show that LDA is optimal, but it is well known that the projection follows a heuristic that often performs well also if the assumption is not met. And which violations are likely to cause trouble: It is also known that heavy tails in the distribution lead to problems with LDA in practice.
Unfortunately, such knowledge rarely makes it into the condensed writing of a paper, so the reader has no clue whether the authors did decide for their model after well considering the properties of the application as well as of the model or whether they just picked whatever model they came across. 

*Sometimes practical approaches (heuristics) evolve that turn out to be very useful from a practical point of view, even if it takes decades until their statistical properties are understood (I'm thinking of PLS).

*The other thing that happens (and should happen more) is that the possible consequences of the violation can be monitored (measured), which allows to decide whether there is a problem or not. For the application, maybe I don't care whether my model optimal as long as it is sufficiently good.
In chemometrics, we have a rather strong focus on prediction. And this offers a very nice escape in case the modeling assumptions are not met: regardless of those assumptions, we can measure whether the model does work well. From a practicioner's point of view, I'd say that you are allowed to do whatever you like during your modeling if you do and report an honest state-of-the-art validation.
For chemometric analysis of spectroscopic data, we're at a point where we don't look at residuals because we know that the models are easily overfit. Instead we look at test data performance (and possibly the difference to training data predicitve performance). 

*There are other situations where while we're not able to predict precisely how much violation of which assumption leads to a breakdown of the model, but we're able to measure consequences of serious violations of the assumption rather directly.
Next example:  the study data I typically deal with is orders of magnitude below the sample sizes that the statistical rules-of-thumb recommend for cases per variate (in order to guarantee stable estimates). But the statistics books typically don't care much about what to do in practice if this assumption cannot be met.  Nor how to measure whether you actually are in trouble in this respect. But: such questions are treated in the more applied disciplines. Turns out, it is often quite easy to directly measure model stability or at least whether your predictions are unstable (read here on CV on resampling validation and model stability). And there are ways to stabilize unstable models (e.g. bagging).

*As an example of the "2nd line of defense" consider resampling validation. The usual and strongest assumption is that all surrogate models are equivalent to a model trained on the whole data set. If this assumption is violated, we get the well-known pessimistic bias. The 2nd line is that at least the surrogate models are equivalent to each other, so we can pool the test results. 

Last but not least, I'd like to encourage the "customer scientists" and the statisticians to speak more with each other. The statistical data analysis IMHO is not something that can be done in a one-way fashion. At some point, each side will need to acquire some knowledge of the other side. I sometimes help "translating" between statisticians and chemists and biologists. A statistician can know that the model needs regularization. But to choose, say, between LASSO and a ridge, they need to know properties of the data that only the chemist, physicist or biologist can know. 
A: Given that CV is populated by statisticians and people who are curious, if not competent, about statistics, I am not surprised about all the answers emphasizing the need to understand the assumptions. I also agree with these answers in principle.
However, when taking account the pressure to publish and the low standard for statistical integrity currently, I have to say that these answers are quite naive. We can tell people what they should do all day long (i.e. check your assumption), but what they will do depends solely on the institutional incentives. The OP himself states that he manages to publish 20 articles without understanding the model's assumption. Given my own experience, I don't find this hard to believe.
Thus I want to play the devil's advocate, directly answering OP's question. This is by no means an answer that promotes "good practice," but it is one that reflects how things are practised with a hint of satire.

Is it worth the extra effort? 

No, if the goal is to publish, it's not worth it to spend all the time understanding the model. Just follow the prevalent model in the literature. That way, 1) your paper will pass reviews more easily, and 2) the risk of being exposed for "statistical incompetence" is small, because exposing you means exposing the entire field, including many senior people.

Is it not likely that the majority of all published results do not respect these assumptions and perhaps have not even assessed them? This is probably a growing issue since databases grow larger every day and there is a notion that the bigger the data, the less important is the assumptions and evaluations.

Yes, it's likely that most published results are not true. The more involved I am in actual research, the more I think it is likely.
A: The short answer is "no." Statistical methods were developed under sets of assumptions that should be met for the results to be valid.  It stands to reason, then, that if the assumptions were not met, the results may not be valid.  Of course, some estimates may still be robust despite violations of model assumptions.  For example, multinomial logit appears to perform well despite violations of the IIA assumption (see Kropko's [2011] dissertation in the reference below).
As scientists, we have an obligation to ensure that the results we put out there are valid, even if the people in the field don't care whether assumptions have been met.  This is because science is built on the assumption that scientists will do things the right way in their pursuit of the facts. We trust our colleagues to check their work before sending it out to the journals.  We trust the referees to competently review a manuscript before it gets published.  We assume that both the researchers and the referees know what they are doing, so that the results in papers that are published in peer-reviewed journals can be trusted.  We know this is not always true in the real world based on the sheer amount of articles in the literature where you end up shaking your head and rolling your eyes at the obviously cherry-picked results in respectable journals ("Jama published this paper?!").
So no, the importance cannot be overstated, especially since people trust you--the expert--to have done your due diligence.  The least you can do is talk about these violations in the "limitations" section of your paper to help people interpret the validity of your results.
Reference
Kropko, J. 2011. New Approaches to Discrete Choice and Time-Series Cross-Section Methodology for Political Research (dissertation).  UNC-Chapel Hill, Chapel Hill, NC.
A: Well, yes, assumptions matter -- if they didn't matter at all, we wouldn't need to make them, would we?
The question is how much they matter -- this varies across procedures and assumptions and what you want to claim about your results (and also how tolerant your audience is of approximation -- even inaccuracy -- in such claims).
So for an example of a situation where an assumption is critical, consider the normality assumption in an F-test of variances; even fairly modest changes in distribution may have fairly dramatic effects on the properties (actual significance level and power) of the procedure. If you claim you're carrying out a test at the 5% level when it's really at the 28% level, you're in some sense doing the same kind of thing as lying about how you conducted your experiments. If you don't think such statistical issues are important, make arguments that don't rely on them. On the other hand, if you want to use the statistical information as support, you can't go about misrepresenting that support. 
In other cases, particular assumptions may be much less critical. If you're estimating the coefficient in a linear regression and you don't care if it's statistically significant and you don't care about efficiency, well, it doesn't necessarily matter if the homoskedasticity assumption holds. But if you want to say it's statistically significant, or show a confidence interval, yes, it certainly can matter.
A: I am trained as a statistician not as a biologist or medical doctor.  But I do quite a bit of medical research (working with biologists and medical doctors), as part of my research I have learned quite a bit about treatment of several different diseases.  Does this mean that if a friend asks me about a disease that I have researched that I can just write them a prescription for a medication that I know is commonly used for that particular disease?  If I were to do this (I don't), then in many cases it would probably work out OK (since a medical doctor would just have prescribed the same medication), but there is always a possibility that they have an allergy/drug interaction/other that a doctor would know to ask about, that I do not and end up causing much more harm than good.  
If you are doing statistics without understanding what you are assuming and what could go wrong (or consulting with a statistician along the way that will look for these things) then you are practicing statistical malpractice.  Most of the time it will probably be OK, but what about the occasion where an important assumption does not hold, but you just ignore it?
I work with some doctors who are reasonably statistically competent and can do much of their own analysis, but they will still run it past me.  Often I confirm that they did the correct thing and that they can do the analysis themselves (and they are generally grateful for the confirmation) but occasionally they will be doing something more complex and when I mention a better approach they will usually turn the analysis over to me or my team, or at least bring me in for a more active role.
So my answer to your title question is "No" we are not exaggerating, rather we should be stressing some things more so that laymen will be more likely to at least double check their procedures/results with a statistician.
Edit
This is an addition based on Adam's comment below (will be a bit long for another comment).
Adam, Thanks for your comment.  The short answer is "I don't know".  I think that progress is being made in improving the statistical quality of articles, but things have moved so quickly in many different ways that it will take a while to catch up and guarentee the quality.  Part of the solution is focusing on the assumptions and the consequences of the violations in intro stats courses.  This is more likely to happen when the classes are taught by statisticians, but needs to happen in all classes. 
Some journals are doing better, but I would like to see a specific statistician reviewer become the standard.  There was an article a few years back (sorry don't have the reference handy, but it was in either JAMA or the New England Journal of Medicine) that showed a higher probability of being published (though not as big a difference as it should be) in JAMA or NEJM if a biostatistican or epidemiologist was one of the co-authors.
An interesting article that came out recently is: http://www.nature.com/news/statistics-p-values-are-just-the-tip-of-the-iceberg-1.17412 which discusses some of the same issues.
A: If you need very advanced statistics it's most likely because your data is a mess, which is the case with most social sciences, not to mention psychology. In those fields where you have good data you need very little statistics. Physics is a very good example. 
Consider this quote from Galileo on his famous gravitational acceleration experiment:

A piece of wooden moulding or scantling, about 12 cubits long, half a
  cubit wide, and three finger-breadths thick, was taken; on its edge
  was cut a channel a little more than one finger in breadth; having
  made this groove very straight, smooth, and polished, and having lined
  it with parchment, also as smooth and polished as possible, we rolled
  along it a hard, smooth, and very round bronze ball. Having placed
  this board in a sloping position, by raising one end some one or two
  cubits above the other, we rolled the ball, as I was just saying,
  along the channel, noting, in a manner presently to be described, the
  time required to make the descent. We repeated this experiment more
  than once in order to measure the time with an accuracy such that the
  deviation between two observations never exceeded one-tenth of a
  pulse-beat. Having performed this operation and having assured
  ourselves of its reliability, we now rolled the ball only one-quarter
  the length of the channel; and having measured the time of its
  descent, we found it precisely one-half of the former. Next we tried
  other distances, compared the time for the whole length with that for
  the half, or with that for two-thirds, or three-fourths, or indeed for
  any fraction; in such experiments, repeated a full hundred times, we
  always found that the spaces traversed were to each other as the
  squares of the times, and this was true for all inclinations of the
  plane, i.e., of the channel, along which we rolled the ball. We also
  observed that the times of descent, for various inclinations of the
  plane, bore to one another precisely that ratio which, as we shall see
  later, the Author had predicted and demonstrated for them.
For the measurement of time, we employed a large vessel of water
  placed in an elevated position; to the bottom of this vessel was
  soldered a pipe of small diameter giving a thin jet of water which we
  collected in a small glass during the time of each descent, whether
  for the whole length of the channel or for part of its length; the
  water thus collected was weighed, after each descent, on a very
  accurate balance; the differences and ratios of these weights gave us
  the differences and ratios of the times, and this with such accuracy
  that although the operation was repeated many, many times, there was
  no appreciable discrepancy in the results.

Note the highlighted by me text. This is what good data is. It comes from a well planned experiment based on a good theory. You don't need statistics to extract what you're interested in. There was no statistics at that time, neither there were computers. The outcome? A pretty fundamental relationship, which still holds, and can be tested at home by a 6th grader.
I stole the quote from this awesome page.
UPDATE:
To @Silverfish comment, here's an example of statistics in experimental particle physics. Pretty basic, huh? Barely over MBA level. Note, how they love $\chi^2$ :) Take that, statisticians!
A: This question seems to be a case of professional integrity.
The problem seems to be that either:
(a) there isn't enough critical assessment of statistical analysis by lay persons or
(b) a case of common knowledge is insufficient to identify statistical error (like a Type 2 error)?
I know enough about my area of expertise to request an experts input when I am near the boundary of that expertise.  I have seen people use things like the F-test (and R-squared in Excel) without sufficient knowledge.
In my experience, the education systems, in our eagerness to promote statistics, have over-simplified the tools and understated the risks / limits.  Is this a common theme that others have experienced and would explain the situation?
A: While Glen_b gave a great answer, I would like to add a couple of cents to that.
One consideration is whether you really want to get the scientific truth, which would require polishing your results and figuring out all the details of whether your approach is defensible, vs. publishing in the "ah well, nobody checks these eigenvalues in my discipline anyway" mode. In other words, you'd have to ask your inner professional conscience whether you are doing the best job you could. Referring to the low statistical literacy and lax statistical practices in your discipline does not make a convincing argument. Reviewers are often at best half-helpful if they come from the same discipline with these lax standards, although some top outlets have explicit initiatives to bring statistical expertise into the review process. 
But even if you are a cynical "publish-or-perish" salami slicer, the other consideration is basically the safety of your research reputation. If your model fails, and you don't know it, you are exposing yourself to the risk of rebuttal by those who can come and drive the ax into the cracks of the model checks with more refined instruments. Granted, the possibility of that appears to be low, as the science community, despite the nominal philosophical requirements of reputability and reproducibility, rarely engages in attempts to reproduce somebody else's research. (I was involved in writing a couple of papers that basically started with, "oh my God, did they really write that?", and offered a critique and a refinement of a peer-reviewed published semi-statistical approach.) However, the failures of statistical analyses, when exposed, often make big and unpleasant splashes.
