try Resetting Firefox:

Reset Firefox preferences to troubleshoot and fix problems

also see: Profile Manager - Create, remove or switch Firefox profiles

That fixes Ctrl+Shift+Tab to go backwards, but Ctrl+Tab still does not work.

See if this helps;

Customizable Shortcuts {web link} Allows to customize Firefox shortcuts.

Do a clean reinstall and delete the Firefox program folder before (re)installing a fresh copy of the current Firefox release.

• Download the full Firefox installer and save the file to the desktop
  https://www.mozilla.org/en-US/firefox/all.html

If possible uninstall your current Firefox version to cleanup the Windows registry and settings in security software.

• Do NOT remove "personal data" when you uninstall your current Firefox version, because this will remove all profile folders and you lose personal data like bookmarks and passwords including data in profiles created by other Firefox versions.

Remove the Firefox program folder before installing that newly downloaded copy of the Firefox installer.

• (32 bit Windows) "C:\Program Files\Mozilla Firefox\"
• (64 bit Windows) "C:\Program Files (x86)\Mozilla Firefox\"

• It is important to delete the Firefox program folder to remove all the files and make sure that there are no problems with files that were leftover after uninstalling.
• http://kb.mozillazine.org/Uninstalling_Firefox

Your bookmarks and other personal data are stored in the Firefox profile folder and won't be affected by an uninstall and (re)install, but make sure NOT to remove personal data when you uninstall Firefox as that will remove all Firefox profile folders and you lose your personal data.

If you keep having problems then also create a new profile.

• http://kb.mozillazine.org/Profile_folder_-_Firefox
• http://kb.mozillazine.org/Profile_backup

• http://kb.mozillazine.org/Standard_diagnostic_-_Firefox#Clean_reinstall

A complete reinstall (Programs Folder was deleted in uninstallation automatically) and a removal of profile folder did not work. Ctrl + Tab is still not working.

Have you tried Customizable Shortcuts?

Yes and it doesn't fix my problem.

Open Customizable Shortcuts, and check that the shortcuts are correct. Also see if anything else is using the same shortcut.

You can also change a shortcut hotkey if you want.