I am using Lumia 920, Facebook app from Microsoft hasn't been giving me any notification since past couple of days.. Before that it used to.. not any more...
You're not alone my friend, almost everyone got an issue with this app. Personally I uninstalled it and now using the web page pinned to my home screen, yes you still got no notification but its WAY better than the app itself. Hope people at MS pay some attention to this as its a major app.
There is a thread on this but what I do is I went to my Facebook settings enabled all people I want to get notification from and my me tile does the rest...no need for 3rd party app.