I have been using Firefox, and reliably updating it at each opportunity, on both my 27-inch iMac, and my MacBook Pro, and on my daughter's MacBook Air, all running the latest OSX versions. Somewhere around version 26 of Firefox, the Firefox update would download normally, and when I selected it to install, Firefox would seem to restart itself after installing the update.
I would then have two Firefox icons in the Dock. I could use Firefox, it seemed, until I wanted to quit it. It would then not quit. At least one of the Firefox icons in the dock would be persistent, and never disappear from the Dock. Despite reporting as "application unresponsive" and having "Force Quit" as a selection for that persistent icon, it was impossible to quit Firefox, and make the icon disappear from the Dock.
When in that state, I also could not select my Mac to restart or shutdown; Firefox was apparently preventing either action. A check with Activity Monitor revealed no running process that appeared to be associated with Firefox. Forcing the Dock to quit and restart did not help to either reveal a Firefox process, or make the persistent icon go away.
At first, I had to follow all Firefox updates with holding down the start button until system shut down, but this is hard on a file system. Eventually, I tried logging in as another user on another account, and then selecting restart or shutdown from that other account. This would work to restart or shutdown the Mac after specifying an Administrator account and password for authentication. After a reboot, Firefox would work fine on all accounts until the next Firefox upgrade got installed.
It seems that a Firefox upgrade fails to properly terminate the old Firefox version, and it gets into a state where it is not running (no process visible using Activity Monitor) and running (icon visible in Dock) at the same time. if a restart or shutdown is ordered with a persistent Firefox icon present, it may then render that user account useless, as no new process can be started (Shutdown/restart pending), nor can the restart or shutdown request be aborted.
I would appreciate a fix for this someday. If it would help to see this behaviour on a video, then I can try to record what happens on my next Firefox update.