If you re-open the tab using:

• right-click any tab in the window > Undo Close Tab
• Ctrl+Shift+t
• History > Recently Closed Tabs

Does Firefox retain the session data (e.g., cookie) for that tab so you can keep using Twitter?

If you close that tab, then Ctrl+t to create a new tab in the same window and open Twitter, does that recognize your logged in status?

Some extensions may clean up behind you, but the ones I heard about aren't instant. To see whether extensions play a role in this issue, could you try Firefox's Safe Mode? That's a standard diagnostic tool to bypass interference by extensions (and some custom settings). More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using

Help > Restart with Add-ons Disabled

In the dialog, click "Start in Safe Mode" (not Reset)

Any difference?

Websites remembering you and automatically log you in is stored in a cookie.

• Create a cookie 'allow' exception to keep such cookies, especially for secure websites and when cookies expire when Firefox is closed.
• Tools > Options > Privacy > Cookies: Exceptions

In case you are using "Clear history when Firefox closes":

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

• 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

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

Clearing cookies will remove all specified (selected) cookies including cookies with an allow exception that you want to keep.