I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths of a paper of mine (and this is very long !). But I have some doubts. Is it really my job as a referee ?
This question is related to Refereeing a Paper but only a few comments were made on that point in the above cited thread (mainly Refereeing a Paper).
I only hope that computer-aided proof checking saves mathematics before it collapses under the weight of decades of irresponsible publishing. Of all disciplines, peer review in mathematics should serve to guarantee nearly absolute confidence in the validity of published results. Many subjects have grown so complex that one can't reasonably expect new people coming to the field to take responsibility for the correctness of all the literature that they might need to quote.
I remember attending a seminar at a famous institute where a young speaker justified a step by citing a paper by a well-known and well-published worker in the field. A very, very famous mathematician in the audience, the recognized leader of the discipline, stopped him and said "I wouldn't believe anything in [so and so]'s papers." A hush went around the room. Later I asked a colleague of the impugned mathematician (a member of the same department and an expert in the same field) about the incident. "Yeah, everyone knows his papers are garbage" he said. I asked why they get published. "No one wants a fight. We publish them and then ignore them."
I don't want this sort of practice to define mathematics in the public mind. I think we should compensate referees for their hard work, and honor solid refereeing nearly as much as we do excellent research. Either that, or fund computer-aided proof checking to the hilt, change the methodology of the subject and get human beings out of the business of vouchsafing the literature.
I regard it as part of my job as a referee. But the amount of time I spend checking proofs really depends on whether the point of the paper is to prove something I already believed but didn't know how to prove (in which case I spend a lot of time) or to tell me something new, in which case I might spend very little time on the proofs and rather focus on deciding how interesting the new facts are.
I agree with the oft-expressed sentiment that it is primarily the responsibility of the author to check for correctness. Editors may well find your assessment of the value of the theorems contained in the paper more useful than your assurances that the proofs are correct.
Someone who is interested enough in the results of a paper to use them is going to be the most likely source of corrections for the proofs. I've found more errors in papers whose results I needed to apply than in papers I've officially refereed.
Edit: I think it's worth responding to David Feldman's answer. I agree with his aim: ensure that mathematics literature is not full of errors. But I think that the refereeing process is not the most efficient means for weeding out inaccuracies. Better for that to happen organically as the consumers of new ideas put them to the test. The arxiv helps quite a bit, by ensuring that ideas are disseminated quickly to those that are likely to appreciate them. Why centralize this process? Furthermore, even with the most conscientious refereeing, mistakes will slip through if the only two people who have read the paper carefully are the author and a reviewer (and if those really are the only two people who have read the paper carefully, then it's probably not a big deal that an error made it through, anyway).
Roy Smith's comment below is a good one: if you don't have time to check the proofs carefully, tell the editor that. S(he) can then make an informed decision about publication. Often I receive a request to referee a paper that I am sure I will find useful at some point in the future, but I don't have time to check carefully all the proofs (real life and other work can get in the way). I could tell the editor to find another referee, or I could do the best I can in the time I have. Sometimes I really think it's better for the mathematical community to choose the latter, since finding someone to referee a paper can sometimes be a real timesink.
Perhaps the disagreement here is a cultural one: some people think of mathematics as an experimental science, some think of it as primarily about finding the right answers, and some think of it as primarily about finding proofs. In the last case it's natural to put a premium on proof checking.
I agree with David: I think that my job as a referee consists mainly in checking that the proofs are correct.
However, checking that the proof is correct can in practice not be done by line-by-line checking. Referees are not computers, nor are the writers of the articles. Rather I have a kind of critical, "falsificationist" approach. I try to see what are the principal steps of the proof. If the article is well-written, this work has been done by the authors. Each of those stepsis an assertion, and I try to disporve it. Is it in contradiction with something I know? Can I find a counter-example? If I can't I begin to believe a little in this one step, so I try to prove it myself. If it's too hard, I read the proof of the argument (if there is one!) and apply th same method th understand and check it.
I more or less agree with Sheikraisinrollbank, but for the sake of argument...
I've always found it slightly hard to fully separate "Is this paper interesting" from "Is this paper correct". It seems like mathematics (especially towards the pure end) is full of interesting, plausible "facts" we don't have proofs of. Isn't part of the point of maths to find rigourous proofs for things? So if I want to claim that a paper is interesting, it seems a priori necessary to have some faith in the paper being correct!
Similarly, the "interest" of a paper, to me, is often bound up in the methods of proofs being used, not just the statements of the theorems (is this proof something which I would have tried if I'd thought hard, or is it completely from left-field?) So I'd want to read the proofs carefully, even if I wasn't "checking" them.
So, if had to advise: Yes, you should read the proofs, closely. But checking every line? Perhaps not, unless something looks very off.
I believe that the referee should check the proof is correct, or at least that the logical deduction of its claimed results from the assumed results it quotes is correct. It probably is unreasonable to expect referees to check the correctness of cited results in a paper- this could easily become a never-ending task. While there is probably a general consensus that responsibility for correctness rests primarily with the author(s), the community needs safeguards to ensure that genuine mistakes by authors are caught before they become assimilated into the body of accepted mathematical knowledge.
However, I also believe that authors who quote previous results in their own proof have a responsibility to ensure as far as possible that they fully understand the results they are quoting, and their correctness. If a quoted result turns out to be wrong, the person who quoted it can't evade their own responsibility by blaming the referee of the paper quoted.
Of course there is a conflict between idealised behaviour, and what is attainable in the "real" world, but in the long run, Mathematics, perhaps above all other human endeavours, strives for an idealized perfection, and abandoning that striving would be fatal for the subject.
I am waiting for a time when high ranked journals will require besides mathematical proofs written for human also alternative proofs that can be checked by computers (written in special language like Coq or Mizar). In this case referee will not have the obligation to check the validity of the proofs (this will be done automatically at the submission stage) but concentrate on evaluation of the importance of obtained results etc. This would save a lot of referee's valuable time and simultaneously will guarantee that the papers accepted for publication do not contain errors.