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 bar.
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. and whatever else you want.

Let the cookies expire when Firefox is closed to make them session cookies instead of using "Clear history when Firefox closes" to clear all cookies including cookies with an allow exception you may want to keep.

• Options/Preferences -> Privacy & Security
  Cookies and Site Data: [ ] "Delete cookies and site data when Firefox is closed"

You can create a cookie allow exception and use "Delete cookies and site data when Firefox is closed" to let cookies without allow exception expire when Firefox is closed.

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an allow exception you may want to keep