I heard that De GiorgiNashMoser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there some cases where the Moser iteration can be successfully applied to elliptic systems?

Hi. The point is not the ellipticity. In fact the Argument of De Giorgi, Moser and Nash was designed for elliptic problems. The point is that solutions $u: \Omega\to\mathbb{R}^N$ of elliptic problems with $N>1$ just aren't $C^{1,\alpha}$ any more in general. This is no problem with the method, it's intrinsic. The famous counterexample is by De Giorgi himself. See Giusti  The direct method of variational calculus for more details to this topic. The counterexample itself can be found as example 6.3 in this book. The paper from De Giorgi is "Un esempio di estremal discontinue per un problema variazionale di tipo ellittico", Boll. U.M.I., 4 (1968), 135137 


Basically, the DeGiorgi  Moser  Nash regularity result fails for elliptic system. As Johannes pointed out, there is a counterexample in Giusti book. For system, usually, one can get "partial regularity result". What I mean by that is: there exists $\Omega_0 \subset \Omega$ open such that $\Omega\setminus \Omega_0=0$ and $u \in C^{1,\alpha}(\Omega_0)$. Then with the smoothness of coefficients, you can have $u \in C^\infty(\Omega_0)$. There are several references you can have a look at are:
In the third one, Evans proved the "partial regularity result" for a minimizer of certain energy functional. The most important thing is that the Lagrangian only need to be uniformly striclyquasiconvex instead of uniformly convex. And it has some very important applications in elasticity. You can search for some papers of John Ball for this issue. 

