The Facebook Messenger app on the phone is very unstable, sad to say. I've had the behavior you describe a few times before on my Lumia 950 XL, and it sucks. It seems to take longer to happen if I always close the app (press-and-hold on Back and then swipe the app to the bottom of the screen to shut it down) than if you just leave it suspended in the background (just hitting the Start or Back buttons to return to the Start screen doesn't kill the app, just suspends it).
You can try the beta version of the app, if you don't mind putting up with possibly-new-bugs in the meantime. It's called Messenger (Beta) and is Win10/W10M only. It will probably get the fix rolled out first, but I can't say when that will actually happen.
In the meantime, you have a few options:
- Every time I've gotten Messenger into this state, it's been fixable (temporarily) by rebooting my phone. There's probably a simpler fix - a single process you'd need to kill or something - but practically speaking rebooting is the only simple option. It's annoying but doesn't take that long.
- If rebooting doesn't fix it, delete (uninstall) the app, and re-install it from the Store.
- Use https://messenger.com/ in your phone's browser. It works fine in Edge on the phone. You just probably won't get notifications when doing anything else.
Hope that helps!