Why do variance reduction methods work well and are there any 'backfire' experiences? I used different variance reduction methods, for instance, common random numbers (CRN), antithetic-variates (AV), control variates (CV) and importance sampling (IS) but still cannot wrap my head around the logic. For instance, we deliberately use correlated values for a process that is supposed to be independent. 
I can see the reasons for some of the methods such as CV and IS but for CRN and AV I am kinda skeptic. Therefore, I cannot think of "don't use in this case" examples.
So, are there a layman terms explanation and some anti-examples?
Edit: Some extra explanation. I know about the mechanical reasons that lead to variance reduction. What I cannot fathom is the legitimacy of manipulating random variate generation with correlated streams which should have been independent otherwise. I sometimes have the feeling, using VR techniques might manipulate the confidence interval to an undesired position.
 A: Your skepticism is well warranted. No matter how attractive they seem, most techniques used for variance reduction will fail in certain corresponding settings, so that variance reduction is more of an art than settled science. One might lose unbiasedness or even consistency of estimators, and even when they are preserved the computational cost might be higher than just drawing more samples in the first place. In general for complex models it is hard to know "in advance" if a variance reduction technique is appropriate, tailored analysis is necessary.
E.g. antithetic variates will waste sampling effort when the function is radially symmetric. 
A control variate too must be cheap and effective enough for the increased effort to be compensated by precision gains, which is not always the case. And it can be used for expectations/integrals while it is not intended for other operations such as quantiles (although it has been used in that context too).
Importance sampling can easily backfire when the sampling measure is not appropriately mimicking the target (e.g. it has too fast tail decay compared to the target, which will lead to wildly varying likelihood ratio weighting).
There are various papers on problems of classical variance reduction approaches used in Quasi MC such as recursive brownian bridging. Lattice rules work best under restrictive assumptions on the integrand. Standard Sobol' sequences can also have other drawbacks and sometimes even lead to biased estimates.
Even apparently simple one-dimensional MC integration (without any curse of dimensionality, and even smooth functions) will be tricky for fat tailed distributions or oscillatory integrands already before any variance reduction, which only complicates issues.
And so on...
You must know well what you're doing and all the limitations of the specific VR method adopted, or you'll risk bias or slow/no convergence, unless your integrand is simple enough.
A: I will address common random numbers (CRN), and leave you to read section 4 of Columbia University IEOR E4703 2004 lectures by Martin Haugh on Variance Reduction Methods I for a discussion of antithetic variates.
Use CRN when you want to compare performance (results) for 2 or more cases. Perform $n$ replications for each of $m$ cases being compared ($m = 2$ is particularly common, used to estimate the difference in performance of 2 cases).  Each case gets fed the same random numbers, which eliminates a confounding source of variability. Basically, CRN eliminates common error, which leaves a smaller remaining error.
If positive correlation is achieved between the cases being compared, variance is reduced.
If you are comparing (differencing) 2 cases and get negative correlation, variance is increased, so it would backfire. Your simulation would have to be pretty screwed up to achieve this. Nevertheless, I regularly encounter people in the real world, some of whom have pertinent graduate degrees, who time and again demonstrate a talent for screwing up even the simplest things so badly that they could pull this off.  But such people screw up everything.
The real challenge with CRN is to be able to maintain a sufficient level of synchronization in the random numbers (how they are used) in order to achieve a non-trivial magnitude positive correlation. Use of multiple random 
number streams within a simulation might help facilitate this to the maximum extent possible.
Here is an example where CRN which achieves positive correlation can be of significant benefit, and where it id quite realistic, if using a properly designed simulation program, to achieve good results using CRN.  Let's say you are estimating the gradient of $Ef(X)$ with respect to the components of $X$.  A central difference estimate of the ith component of the gradient = $$\frac{Ef(X + h e_i) - Ef(X - h e_i)}{2h}$$ where $e_i$ is the ith column of the identity matrix.  
If $Ef(X + h e_i)$ and $Ef(X - h e_i)$ are evaluated independently, and have a common variance $V$, the variance of the difference $= 2V$, which then gets multiplied by $1/4h^2$, which will typically be very large, because $h$ needs to be small to keep truncation error small enough. 
If, on the other hand,  $Ef(X + h e_i)$ and $Ef(X - h e_i)$ are perfectly correlated, the variance of the central difference estimate is $0$.  On realistic problems, a positive correlation $ < 1$ might be achieved, resulting in some (possibly quite considerable) variance reduction, but not variance elimination. 
If you really screwed things up and achieved a correlation of $-1$, the variance of the difference would be $4V$, which is double the variance of the independent case.  If I ever needed to achieve this, I would give it to one of the many people I have encountered who screw up everything. Unfortunately, they couldn't be counted on to perfectly screw this up, so even that might not work.
EDIT in response to OP's edit:
Independent Sampling: Estimate $A$ and $B$ independently. Then form a confidence interval for $A$ and $B$ based on $A$ and $B$ being independent. 
Use of Common Random Numbers (CRN): Directly form confidence interval for $Z$, defined as being $A - B$.  I.e., estimate a confidence interval for $Z$, which has n data points, the ith data point of which is $A_i − B_i$.  There's nothing illegitimate about that. All that's being done is to form a confidence interval for $Z$. If $A$ and $B$ are positively correlated, then $Var(Z) < Var(A) + Var(B)$. Note that I have used estimates in place of the variables in certain places in the write up, in a way which should be obvious..
