Edit: Though the fix is the same, echobucket beat me to it in the other thread here:
http://discussions.apple.com/thread.jspa?threadID=979688&tstart=0
This might not work for you, but after a while poking around on my HD to try to find the cause a few days ago, this killed the autocheck and it's associated password requirement somewhat nicely.
1. Quit iTunes.
2. Open your iTunes folder (usually found in the Music folder in your Home folder).
3. Double click the iTunes Music folder.
4. Inside you should find a folder called Downloads.
5. Drag the Downloads folder to your desktop.
6. Open iTunes and hopefully bask in the glow of it not asking for your password.
The folder goes on the desktop just to make sure that it works, and so if moving it makes iTunes cry/break/explode in some way, it can always be dragged back. If iTunes works happily and all your purchased music has been downloaded successfully, it's probably safe to delete it.
Just in case anyone is wondering, iTunes is asking for your password to check for purchased (but undownloaded) music, and appears to be a bug which occurs when you quit iTunes with a download in progress. The folder's purpose seems to be to remind iTunes to check for purchased but not yet downloaded music, and for whatever reason, iTunes doesn't seem to be killing said folder when the music has been downloaded successfully finishes.
Filed as a bug.
MacBook Pro Mac OS X (10.4.9)