I've got 10.10.10 over here and also since I upgraded to 3.6.23 I got exactly the same issue. When I tried to shut down, a message box prompted me that firefox-bin wasn't shut down proparly and I had to kill the process or force a shut down.
Also when I closed Firefox and I tried to start it again it prompted that Firefox was already running.
The "lock" file seen to be a hyper link to an non existing target (broken).
After I've deleted this and the ".parentlock" file, Firefox did close down properly again and could start it as normal after I've closed it.