I know there was some sort of Facebook and AT&T issue where others' notifications showed up, but I don't know if the iPhone was included in the list of affected phones. You can try looking at the Facebook Settings (Settings app, towards the bottom). In the Facebook Settings, choose the Push Notifications option. (This is different than the Notifications option in the main settings screen.)
If this doesn't work, I would personally uninstall Facebook, reboot the device (yes, I know this is not supposed to be necessary, but I'm old school Windows), and then reinstall it from the app store.