I recently found out that bioRxiv, the main biology preprint server, actually does not allow postprints to be submitted (even if allowed by the journal) according to its policy. This is in contrast to arXiv (after which biorxiv was modelled) which does allow submission of postprints.
To clarify, preprints are the authors' version of the paper prior to intial submission, and postprints are the authors' version of the paper which was accepted for publication (typically without the editing performed by the journal). The term "postprint" is a little confusing because the paper will often be accepted by the journal but not printed or released yet.
I can understand that these two types of manuscript are different and should probably be tagged differently, but I can't really see a significant benefit in not allowing postprint submission - after all they do provide access to otherwise paywalled information, often before it is actually published.
I would be interested to know if there are any alternatives to bioRxiv which allow postprint submission.