Your ‘logged-in’ status is stored in special cookies. If the cookie is removed, you are logged out.

You may be in Private Browsing mode or told Firefox not to remember.

Type about:preferences#privacy<enter> in the address box. The button next to History, select Use Custom Settings.

Turn off Always Use Private Browsing Mode Turn on Remember My Browsing And Download History At the bottom of the page, turn on Clear History When Firefox Closes. At the far right, press the Settings button. Turn on ONLY Cache and Form And Search History leaving the others off.

Macs: Make sure you install Firefox properly and that you do not run Firefox from within the DMG (Disk Images) file. https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac

Just tried those settings and unfortunately that did not work.

Many site issues can be caused by corrupt cookies or cache.

Warning ! ! This will log you out of sites you're logged in to. You may also lose any settings for that website.

• Delete cookies to remove the information websites have stored on your computer {web link}

• How to clear the Firefox cache {web link}

If there is still a problem,

Start Firefox using Troubleshoot(Safe) Mode {web link}

A small dialog should appear. Click Start In Troubleshoot(Safe) Mode (not Refresh). Is the problem still there?

While you are in Troubleshoot(Safe) mode;

Try disabling graphics hardware acceleration in Firefox. Since this feature was added to Firefox it has gradually improved but there are still a few glitches.

How to disable Hardware Acceleration {web link}

Just tried your 4 steps and still not working.

Note that on Mac closing Firefox can be more complicated because you can close all open windows, but keep Firefox running with the menu bar visible or minimized in the Doc.

Make sure to use "Quit Firefox" with all windows still open, especially if you want to use "Restore Previous Session". Otherwise check "History -> Restore Previous Session" and "History -> Recently Closed Tabs/Windows".

Detail like websites remembering you and log you in automatically is stored in a cookie, so make sure you aren't clearing important cookies.

You can use these steps to make a website recognize and remember you.

• create a cookie allow exception with the proper protocol (https:// or http://) to make a website remember you

You can check that you aren't clearing important cookies.

• using "Delete cookies and site data when Firefox is closed" keeps cookies with an allow exception
• using "Clear history when Firefox closes" to clear cookies clears all cookies including cookies with an allow exception you may want to keep
• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exceptions for passwords and other website specific data

• Settings -> Privacy & Security
  Cookies and Site Data: "Manage Exceptions"

It's only Google that I'm having an issue with. I remained logged into all of my other sites.

Profile data can be removed or not recorded in several ways;

• 1) Running Firefox in Private Mode (nothing is recorded)
• 2) Privacy settings; (In the address box, type about:preferences#privacy<enter> in the address box.
• 3) Many disk cleaners and anti-virus type programs can remove data from browsers
  Open their settings and have them leave the browsers alone.
• 4) Mal-ware.

Profile data can be removed or not recorded in several ways;

• 1) Running Firefox in Private Mode (nothing is recorded)
• 2) Privacy settings; (In the address box, type about:preferences#privacy<enter> in the address box.
• 3) Many disk cleaners and anti-virus type programs can remove data from browsers
  Open their settings and have them leave the browsers alone.
• 4) Mal-ware.

I don't have any disk cleaner or anti-virus and I'm definitely not running Firefox in Private Mode.

Here is a last resort option.

If clearing cookies didn't help then it is possible that the cookies.sqlite file that stores the cookies got corrupted.

Rename (or delete) cookies.sqlite (cookies.sqlite.old) and delete other present cookies files like cookies.sqlite-journal in the Firefox profile folder in case the file cookies.sqlite got corrupted.

http://kb.mozillazine.org/Cookies

https://support.mozilla.org/kb/Deleting+cookies