If you can't log out (logoff) then you'll need to delete the "BYC cookies" which keep you logged into the forum. The only thing that keeps you logged into BYC (or anywhere) is your internet browser cookies.
The best thing to do would be to delete just the cookie for BYC. That can be a little tricky, so usually people will dump / delete all their cookies. Keep in mind that this will remove your "auto login" to places besides BYC.
Once you come back to BYC you should be logged out. If not, then there is a problem on your PC or with your ISP. Again, the only way BYC can know who you are and to keep you logged in is by the cookies on your computer.
You could also make sure the option in your profile to always keep you logged in is unchecked.
If cookies are cleared out completely (and maybe even your browser cache) then there is something odd going on local to your PC. You may also want to check if you have any optimization / security programs running on that PC.
If you are having problems staying logged in or problems logging off the forum it is 99% of the time a problem with your internet cookies. You can navigate away from the site, delete the specific site cookies (if you know how) or delete all of them, but do so in this process:
1) Close all your browser windows.
2) Delete the cookie(s)
3) Re-open the browser and navigate back to the site.
Yup, agreed! If you are savvy you can navigate to your cookies directory and just delete the ones for the forum. Just make sure your browser windows are closed so your cookies aren't cached.