Yes, but they can still log in under the new app id, without having to register again. Facebook will just push a popup to re-aurthorize the app again, same way it does if you de-authorize it on facebook.
As far as I'm aware, Facebook now provides app-scoped user IDs; the same FB user will have different user IDs given to two apps (which prevents them from sharing information). So I'm not sure that's the case any more unfortunately.