Firefox for Mac can get stuck in an update loop if you have been using an older version and it’s downloaded a new version to install, but before it gets to install it, you manually update to a new version.
The symptom is that it gets stuck with a dialog box saying “Firefox is installing your updates and will start in a few moments”
You can force-quit the updater app, but whenever you launch Firefox, it comes up again.
Deleting everything related to Firefox or Mozilla in the ~/Library/Application Support folder doesn’t fix this, nor does deleting preferences.
What you actually need to do is go into ~/Library/Caches and delete the Mozilla and the Firefox folders (you may not need to delete both of them, but I didn’t try just one).
Then relaunch Firefox and it should start up as normal.