Firefox will not store cookies for longer than a session.
A likely cause is the file that stores cookies may be corrupt.
1) Close all instances of Firefox.
2) Click the Windows Start button, and type %APPDATA%\Mozilla\Firefox\Profiles\ in the Start Search box near the bottom of the Start menu, without pressing Enter. A list of profiles will appear in the top left of the Start menu (this applies to Vista. For XP and Windows 2000 view: http://support.mozilla.com/en-US/kb/Profiles#On_Windows_2000_and_XP).
3) Click on any of the profiles (e.g. xxxxxxxx.default) to open it with Windows Explorer.
4) Rename cookies.sqlite file to cookies.tmp.
5) Restart Firefox.
6) Log into your sites and see if Firefox remembers the logins.