Can I publish the reviews I write?
The standard rule in my community is that once I finish reviewing a paper, I'm supposed to pretend that I don't know the paper exists.  In particular, I am not supposed to use any insights I gained by reading that paper in my own research.  I am not supposed to reveal the results to my colleagues.  Some venues ask that I destroy any copies of the paper I'm reviewing, along with any programs or data I used to verify the paper's results.  This embargo lifts only when the paper is finally published, but I am never supposed to reveal my identity to the authors, even indirectly.

Under those rules, publishing reviews is completely unethical.  Maybe it would be okay if I had the explicit permission of both the authors and the editor, but I would expect most authors and reviewers to vehemently object.  I would feel weird even asking.

But even under less stringent reviewing rules, I think posting reviews is a very bad idea.  Criticism is best given privately.  One of the purposes of anonymous reviewing is to give authors brutally honest feedback on their work.  Referees can offer direct criticism without worrying that it will harm the authors, and authors are more willing to hear that criticism because they know it will never be public.

Yes, that means authors sometimes get credit for ideas that I suggest in referee reports.  (Most authors are nice and thank the anonymous referee.)  On the other hand, several referees have offered suggestions that have significantly improved my papers, so it all comes out in the wash.

Update: I should add that these ethical constraints attach only to reviews of unpublished papers being considered for publication.  Once a paper is actually published, everyone is free, if not encouraged, to publish their own reviews of the published version.