Do you mean names and passwords in the Password Manager or do you mean that you are no longer logged on to (remembered by) websites after closing and restarting Firefox?
There is a difference between remembering the name and password in the Password Manager and a "remember me" check box on a web page.
The latter usually involves the creation of a special "remember me" cookie that is stored on your computer and that is send to the server.
If that "remember me" cookie is removed or gets corrupted or is not send to the server then the server won't remember you and you need to sign in once again.

• http://kb.mozillazine.org/Cookies
• http://kb.mozillazine.org/Password_Manager

I meant names & passwords in Password Manager. I followed the second link you (cor-el) posted, and links therein to an article about losing passwords, and found what I think is happening: I had checked under Privacy settings, Clear History when exiting FF, and under Settings, Saved Passwords. Since I rarely exit FF except for restarts on updates, I was attributing the loss of passwords to the update, rather than the restart. I tend to think of the browsing history independent of the stored passwords, so I didn't suspect that Privacy checkbox - and I don't think I had ever looked under the Settings. Thanks for the post!