Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples? Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples?
I feel like it is true but not sure exactly how to prove that...
 A: Next to the simple and effective example of Ben, here is a more applied and specific one: Imagine you try to estimate a causal effect, but your regression model to estimate this causal parameter is misspecified (often called "omitted variable bias").
A well-known example deals with returns to schooling, i.e., how much more you earn due to (i.e., in the sense of a causal effect of sitting in class, taking courses) additional schooling. If you simply regress earnings on schooling the regression is likely misspecified, because people (after compulsory schooling ends) choose how much schooling they want. Now, more motivated and able students will, as a tendency, find the idea of going to school for longer less daunting than other students. Now, such able and motivated persons will however also be likely to be good at the workplace due to these characteristics, irrespective of how much schooling they have. Hence, they will likely earn more.
Hence, you would need to control for things like ability/motivation - which  may not be easy in practice - in your regression (and likely other things, too).
Just collecting more data on your simple regression of earnings on schooling will, in turn, not save you from this problem, so both biased and inconsistent estimation. For both small and large datasets, the simple regression, as a tendency, compares earnings of students who are both able and have higher schooling to earnings of students who are less able and have less schooling. Assigning the entire difference in earnings to schooling hence will overstate the causal effect of schooling.
A: Consider the estimator $\hat{\theta} = 3$.  If this estimator is estimating a parameter that is not equal to three then it is biased in all finite samples.  Is this estimator asymptotically consistent?
A: Yes, in some circumstances, no in others. For example, if a bias results from a self-inconsistent assumption, then no. Examples of this latter include omitted variable bias and AIC in the case of censored data, which violates the maximum likelihood assumption. Examples of when it pertains would be AIC in the case of complete support (i.e., without censoring such that the maximum likelihood assumption pertains), and ordinary least squares for equidistant independent axis data. In still others, for example, variance is generally unbiased, but standard deviation is not, see this. Standard deviation would still be asymptotically correct because the small number bias would reduce to zero for $n\to\infty$. Nevertheless, one should not rely on just any asymptotic convergence, if a rather better estimator is available, and see how this was done in this example.  Briefly, if you small number correct standard deviations from a large number of 2 sample SD's and then average them, you will obtain a more variable estimate than if you root mean square combine all the variances and then use a much lesser small number correction for the total number of trials. Some people are surprised at how ineffectual  AIC can be for small samples. Thus, how fast asymptotic convergence occurs can be critical to interpretation of statistical results, and sometimes, for example for AIC, when we do not have measures that inform us of how precise or accurate statistical results are, it can be problematic.
Thus, the question of whether or not a procedure is asymptotically convergent is not by itself a sufficient criterion of validity of statistical results. We also need confidence intervals for those results.
A: 
Is it true that an estimator will always asymptotically be consistent
if it is biased in finite samples?

The correct reply is trivial and it is NO, as pointed out above.
However your question immediately suggest a more interesting one:
Is it true that an estimator will always asymptotically be consistent if it is unbiased in finite samples?
The reply is: yes, if its variance going to zero when sample size diverge.
I add this part here because I suppose can be interesting for some readers.
