Mysterious. Apart from the fact that Mobile Data can be disabled in individual apps, there are no other features that prevent an app from not being able to be used with Mobile Data.
Try this. Go in and disable the use of Mobile Data for the Facebook app. Then open the App Store and check for an update - after that restart your iPhone. Once it is on, go back and enable Mobile Data for Facebook, and see if that made a difference. If this doesn't work, try deleting the Facebook app, restarting and reinstalling it.
Another thing. If you have an active VPN connection, try disabling it and see if that helps.