As per my comment, this was actually asked in the NY Times Science section.
Q. If I tweeze out the single hair that grows from a facial mole, will it cause cancer, as my friend insists?
A. “No,” said Dr. Mark D. Kaufmann, a dermatologist in Manhattan. “Even by cutting, you don’t change the behavior of a mole.”
Netdoctor (a UK site), also agrees with this (emphasis mine):
There is no evidence to suggest that removing hairs from moles is likely to be harmful in any way. Most moles are not 'cancerous' and do not have the potential to become so.
Boston.com (an extension of the Boston Globe?) also says the same thing as the NY Times article:
No. ''You can do anything you want to a mole, pretty much," said Dr. Bernard Cohen, interim chair of dermatology at the Johns Hopkins School of Medicine. ''If you want to pluck, shave, wax, or use electrolysis on a mole, there's no evidence that this will cause a melanoma, or any other kind of skin cancer."
As a matter of fact, virtually every site said the same thing when I searched google trying to find people who actually believe that it does cause cancer to remove hair. So I would say that the studies have been done, and overwhelmingly concluded that hair removal has no effect. Most of us here are not privy to the published Journals though to cite them directly. I tried to refine my search to just .edu sites, and couldn't get to the primary documentation though. (For instance, Google Scholar turned up this one for AU$4. I suspect that this publication may also address the concern, but it doesn't clearly state so in the abstract.)
This may be a good opportunity though to apply some scientific type of thought to the question. We know what melanoma is, so how would the process of removing a hair cause a benign cell to become cancerous? There is a possible mechanism (thanks Sean) via irritation, however it doesn't seem conclusive.