Should I post my PhD Thesis on arXiv, if I intend to make a paper based on it? I am interested in posting my math Ph.D. thesis on the arXiv, but I also plan on writing a shorter paper based on some of the results in my thesis.  I plan on posting this paper on the arXiv when I am finished.

Would this be appropriate?
I'll offer a counterpoint to the answers which advise against posting to arXiv.  I have posted my PhD thesis (in math) to arXiv, as well as the papers that I published from it.  I feel good about this decision and haven't had any problems, and I disagree with most of the objections that have been raised.

Here are some reasons why I think it can be advantageous to post:


Convenience of access.  Sure, my PhD university does have an online thesis repository that contains mine, but it isn't so easy to find and access.  I just tried, starting from a Google search, and it took me at least five minutes to get to the PDF (the first few hits all led me to an alternate paywalled repository).  You might like to try finding my thesis in this repository for yourself, starting from scratch, and see how long it takes you.  (Post your time in comments!)  As an added handicap, you don't know off the top of your head the name of the university; and my username below is my real name, but not my full name.   Then see how long it takes you to find it on arXiv.

Of course, the thesis is also on my professional website (well, the arXiv link is there), but we all know how those tend to move around and vanish.  It'd be worse if I had a common name.  So for the challenge, assume it isn't there.
Archival.  In 300 years, will my university's repository still exist?  I guess the data will still be around somewhere, but will it be in a location that people can readily find, and in a format that they can use? (Brain implant uploads?)  The university may not exist anymore; for that matter, the country where it was located may not exist anymore.  I have quite a bit more confidence that arXiv will survive into the future.  I'm not really arrogant enough to think that people will be clamoring to read my thesis in 300 years, nor even 3, but I wrote it as a permanent document and I'd like it to be one.
Updates.  In the arXiv version of my thesis, I've included an addendum correcting some errors and mentioning some additional citations that I've discovered since graduating.  I can't do that for the institutional repository version.


Some other remarks:


My thesis has a fair amount of additional expository material that isn't in the corresponding published papers, so I think there is some added value in having it available.  This might not apply if you have a "stapler" thesis.
I posted the thesis on arXiv some years after the papers were published (simply because I never thought of doing so until then), so there wasn't really any risk of it "overshadowing" the papers.  If you're concerned about that, you could also wait to arXiv the thesis.  In the short term, your professional web page should be a reasonably good way to find it.
My thesis didn't get flagged as having "overlap" with the papers, which are also on arXiv, although there is a lot of overlap.  Anyway, if it had, the fact that I posted the thesis later should mean that it would be the thesis that would have been flagged, not the papers; and I think people would understand why.
Coauthors: I agree that if your thesis contains joint work, that would be a reason to hesitate.  But I would downgrade Anonymous Mathematician's "really unfriendly" to "ask your coauthors for permission".  And make sure it is clear that the thesis contains joint work and that the coauthors are credited.  I would probably list them in the arXiv comments field, and maybe put an extra page at the start of the thesis PDF with the same info.
The thesis is less polished than a paper: True, but I also have on arXiv my undergraduate thesis and some lecture notes which are even less polished.  I guess it's more important to me to have these writings available, in case someone finds them useful, than to reserve arXiv for only my very best work.
arXiv has hosting costs: I don't really understand this objection, since the marginal costs of hosting one more document are surely extremely small.  But you can always make them a donation if you want to help defray those costs.  (Based on a quick back-of-the-envelope calculation, I estimate that a one-time gift of US $0.30 is sufficient to cover the storage and hosting costs in perpetuity.)