Ok so have you tried removing the account and adding it back or uninstalling the app and reinstalling it? It could be just a one time glitch and setting it up again may fix the problem.
Hmm very unusual and one of the reasons I won't use the app is because there is no option to logout so its running all the time. You on the other hand are getting prompted every time. I suspect it's an app issue or something wrong with the authentication server. Short of another forum users suggestion it might pay to post this question on the FaceBook forum. I think if it was my device I would try a reset however I don't know if you want to go to that extreme.