Perhaps then, the issue is that Firefox is simply not creating the cookie when you log in, instead of deleting the cookie when you close the browser.
My next step in troubleshooting Firefox is to create a new profile and see if the error continues to exist. Here is the official documentation for this feature. You can do this by (on Windows), starting up Firefox in profile mode. Make sure Firefox is closed to start with. Go to Start, Run, then type without the quotes “firefox.exe -p” with the dash-p at the end. This will open up the Profile manager. Click the “create profile” button, hit Next until you are back at the Profile manager, then click the Start Firefox button. Then try and reproduce the error – go to a login page, login, close Firefox, open it again [it opens to the last used profile by default, but you can manually ensure you are loading the new profile by using Run, firefox.exe -p, then skipping the steps to create a profile and click the Start Firefox button].
If your brand new profile remembers your password, then we know it’s an issue with your other Firefox profile, and the best solution is to export your bookmarks, delete the old profile, and use a blank one.
If your brand new profile does not remember your password, there is something seriously fishy going on. Try logging in and closing the window in another browser, IE or Google Chrome.