EDIT: Not an advertising issue, as pointed out throughout this thread. Potential solution (works for me) is to change the forum theme. Go to the lower left of the forums and use the drop-down box. Change to the last option XDA 2013 beta 1024. This has fixed the issue for me, as well as for Formicae (post #4) who reported this fix.
Yesterday I started having issues with the website. Whenever I'd visit, I'd have to log back in. However, navigating to a subforum would automatically log me in. I initially assumed that it had to do with me getting Chrome 23 last night. However, then I noticed it happening on my phone's browser (Stock touchwiz browser). I didn't just have two systems go loopy at the same time.
I'm one of those oddballs that uses an adblocker but whitelists domains that I trust. I want you guys to get advertising revenue from me because I support the website. But, I noticed the problem resolved on my phone whenever I got around the popup ad. So, I re-enabled ad-blocking on this domain. Guess what? Problem solved. No more login issues.
Might want to look at your advertisements and see what's preventing users from remaining logged in. I could login and navigate to the subforums, but then if I clicked the forum button at the top, I'd be logged out again until I navigated elsewhere. There's an advertisement doing this.