There is no way to exit the browser without leaving it running in background
Unlike other mobile browsers on Android (Xscope, Skyfire, Miren) there isn't a way to close the browser. You have to hit the back softkey all the way to the beginning of the browser session. How do I fully close the browser without resorting to opening the task manager to kill the process?
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Just like the built-in Android Browser (and other popular apps like Gmail, Maps, Twitter, etc.), Firefox is designed not to need a "Quit" button. When running in the background it goes into a mode where it uses minimal resources. If other apps need more memory, Android will automatically close Firefox.
If you do want to close the browser yourself, this add-on will add a Quit button to the menu: https://addons.mozilla.org/mobile/addon/quit-fennec/
There is now a dedicated Quit button
Menu > More > Quit