I echo the answer that you are not expected to check the proofs as a MathSciNet reviewer. That said, I personally choose to do so, because I believe that this leads to better reviews. When I read a review I want it to tell me what to expect from the paper, and if there is a gaping hole I want to know that fact before I spend hours stuck on it. Of course, this is assuming that the paper is of somewhat moderate to small length, which has often been the case in the papers I've reviewed. I also feel like putting my name on a review gives it my stamp of approval, and for me that stamp means something more than just looking over a paper quickly.

It might surprise you that I've found many errors in papers I've been asked to review. Some unfixable and central to the paper, some small and unfixable, lots of minor typos (which I don't mention in my reviews unless there are just gobs of them), and in one case an error that was quite subtle and it took a good week for me to find a fix. (I added the fix to my review, but only at the recommendation of the author of the paper.) Of course, you better be sure there is an error (perhaps by communicating with the author if appropriate) before mentioning it in a review. In all these cases, I hope that the people who read my reviews find them helpful.