Try clearing your Cookies for the affected websites.
Tools > Options -> Privacy - Cookies = Show Cookies button.
Enter the domain in the Search bar at the top and all the Cookies for that URL will appear. Unless you can figure out which Cookie is for "remember me", you'll need to delete them all.

Hold the {Ctrl} key while you click on each Cookie in the small window. When that list is all highlighted, click the Remove Cookie button in the lower left. When you are done click Close.

If that problem persists, you might need to delete the cookies.sqlite file in your Profile folder.
http://support.mozilla.com/en-us/kb/Profiles#How_to_find_your_profile
You might need to unhide your "Hidden files and folders" thru Folder Options in the Windows Control Panel

Thanks for your reply, the-edmeister.

It seems not to be storing cookies at all. If I visit a few sites and log in, close my Firefox and open it again - no cookies are there.

I deleted the cookies.sqlite file but this hasn't fixed it.

Again, 'accept cookies' is ticked so it's not the obvious!

Any ideas??

Sorted it!

I had ticked the option to clear history when Firefox closes.

I didn't realise this includes cookies but have now unticked that and it all works fine.

Thank you for your help and sorry for being daft...........

What if you can't even view the settings in the privacy tab? A customer of mine is not able to view that portion of the Options.

To see all the History and Cookie settings, select: Tools > Options > Privacy > History: Firefox will: "Use custom settings for history"

See Settings for privacy, browsing history and do-not-track

See also:

• Cookies - Information that websites store on your computer
• Enhanced Tracking Protection in Firefox for desktop