Well? Does it?
If you use cookies and are on a shared or public computer, not logging out can post a security risk, as someone could browse and post on the Board as you.
As far as server issues…this would depend on the specifics of how jdavis has the server set up to deal with persistent http connections. My best guess is that it is set up like most vBulletin Boards, in that it doesn’t hurt anything to stay logged on. But am I happy to be corrected if I am wrong.