I've seen at lot of threads regarding this issue. Some reinstall the app and relink to Facebook and then it works,etc etc.
Not for me though and I'm suspecting my Windows Phone 8 (Lumia 920)
So I did these steps yesterday:
Deleted the people/Mail/contacts app, went into account.live.com and deleted the connection to Facebook.
Reinstalled the apps and added the connection again.
And it worked - for maybe an hour. After that the "Connected to FB icon" disappeared. Back to square one.
So I tried it again, but this time I deleted the Facebook connection in my WP8 first. Then I did the above steps again.
And now it worked. For a whole day. So, I was happy for awhile there. But of course I wanted my Facebook contacts in my WP8 so I added the Facebook connection on the phone again. As soon as I clicked "Connect" (I saw this "live" ;)) my Windows 8 computer dropped the Facebook connection. :(
And yes, I use the same Live account on my Windows 8 computer, WP8 and Xbox.
Someone else who has the problem, are you also have a WP8 using the same Live account?