Okay. Several hours ago I cleared my cookies, shut down Firefox, then restarted and logged back in to the Forums. When I came back now I was logged in already, "remembered" from earlier. So, at least in my case, clearing the cookies seemed to have fixed it.

To the best of my knowledge, this was the first time this has happened since the last time I had to change my password. It makes sense to me that the cookie was not properly updated when I did so, and was therefore keeping the site from remembering me.

The only suggestion I can make to those who still have the problems is to check how your browser is handling cookies. I know that, with Firefox or Netscape you can set your browser to accept "Session Cookies" which I think are supposed to be deleted when you close your browser. It's a little more complicated in Internet Explorer (duh!), but I think something similar happens. For sites like this, where you want to remain logged in all the time, you have to accept cookies completely, not just for the session.