I had this problem a while back. It was due to a corrupt cookie file.
The cookie file is located is %appdata%/Mozilla/Firefox/Profiles/*your profile*/cookies.sqlite
The easy way to fix is to simply delete the cookie file, a new one will be created when you next start firefox. Obviously you lose all cookies and you have to log in to all your sites again etc. But it should now remember the log in's.
You can also try to salvage the non-corrupt cookies from the file. Here is a guide how:
How to recover a corrupted Firefox cookies.sqlite ยป ghidinelli.com