How accurate are published papers?
No. The publisher does not and cannot guarantee the correctness of the papers. Also the peer-review system is not perfect.
That being said, even though a single paper might have some chance of being wrong, as studies are replicated, and follow up studies verify and extend the conclusions, the scientific community can build up stronger claims.
One paper specifically looked at Why Most Published Research Findings Are False and stimulated a lot of discussion around the topic.
EDIT to provide more context, as requested in comments.
The paper (published in PLOS Medicine, which defines its scope) uses statistical methods to examine the probabilities that specific study types in biomedical research will yield significant results that end up being published.
Some quotes:
the high rate of nonreplication (lack of confirmation) of research discoveries is a consequence of the convenient, yet ill-founded strategy of claiming conclusive research findings solely on the basis of a single study assessed by formal statistical significance, typically for a p-value less than 0.05.
most research questions are addressed by many teams, and it is misleading to emphasize the statistically significant findings of any single team. What matters is the totality of the evidence. Diminishing bias through enhanced research standards and curtailing of prejudices may also help. However, this may require a change in scientific mentality that might be difficult to achieve.
Despite a large statistical literature for multiple testing corrections [37], usually it is impossible to decipher how much data dredging by the reporting authors or other research teams has preceded a reported research finding. Even if determining this were feasible, this would not inform us about the pre-study odds.
While some details of the methodology have been disputed, the gist of the article – that replication of studies is crucial to establish facts, and that the current publication system is biased against replicable research in many ways – has been widely acknowledged well beyond biomedicine, and the paper has triggered a wave of studies examining this range of systemic problems that the paper highlighted.
Publication in a peer reviewed journal is only the start of the road to the acceptance of an idea by the research community, not the last. Peer review should really only be regarded as a basic sanity check of the paper and not relied upon. Once the findings of a paper have been replicated, or used as the basis for further research, then we can be more confident that the paper is correct, but it is still the readers responsibility to make sure they judge the paper for themselves. If a paper has been cited a large number of times (without the citations being refutations! ;o), then that suggest the work is probably sound. Uncritically trusting a paper because it has been peer-reviewed is essentially an example of the "argument from authority" fallacy (in this case the anonymous reviewers being the authority). While peer-reviewed papers are more likely to be correct than, say blogs, there can be no guarantee of this, and it is best to maintain a skeptical attitude.