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.

Do not use Clear Recent History to clear the "Cookies" and the "Site Preferences"

• https://support.mozilla.com/kb/Clear+Recent+History

Clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, and passwords.

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

Say I just booted up my PC. I go to Tools\Options\Security\Passwords and NO sites, user ID's or passwords are listed. Then I go to a web site log in page, I need to enter my password because the field is blank. My user name is showing because I previously checked "remember my user ID" on the log in page. After I click "log in" a Firefox popup comes that says "Remember password for "my user name" at www.ThatWebSite.com." I click on the box in that pop up that says "Remember password" and off I go to the web site. If I then go to Tools\Options\Security\Passwords, that web site, user ID and password are shown. If I shut off my PC and reboot, that information disappears and I have to enter my password again at the same web site where Firefox just saved my password. I have Tools\Options\Privacy\History is set at "Remember History"

Make sure that you do not use Clear History to clear the Saved Passwords and other possible items when you close Firefox.

• Tools > Options > Privacy > Firefox will: "Use custom settings for history" > [X] Clear history when Firefox closes > Settings

• https://support.mozilla.org/kb/Clear+Recent+History

If you use cleanup software like CCleaner then check the settings for the Firefox application.

I do not clear recent history. The saved passwords disappear after a re-start and may disappear during a session of web surfing. I will check that more concerning during a session. Maybe it's malware? I use StopZilla anti-spyware and the free version of Avast anti virus.

Start Firefox in Diagnose Firefox issues using Troubleshoot Mode to check if one of the extensions or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox (Tools) > Add-ons > Appearance/Themes).

• Don't make any changes on the Safe mode start window.
• https://support.mozilla.org/kb/Safe+Mode