Also happens on startup for me and others. See for example this comment from the bug report:
I simply double-clicked the Firefox icon twice quicker than I ever normally would, and the Close Firefox error appears: "Firefox is already running, but is not responding. To open a new window, you must first close the existing Firefox process, or restart your system."
This is actually due to shutdown (rather than startup) being too slow. Your profile is still in use from the instance of Firefox taking too long to shut down, so when you start a new instance it hits this error. This should be a little better with multiprocess, because web pages are run in a separate process, and we kill that process more quickly, so shutdown should be faster.