There has been an issue of people being unable to delete a Facebook account from the iPad settings, and being continuously prompted to enter a password.
Per the Apple discussions page, there does not seem to be a fix short of wiping the iPad.
I had the same issue — I had an old facebook account in the iPad integration settings and kept getting errors re. automatic login. I deleted the Facebook app thinking that would fix it … but it didn’t.
Finally, in Settings / Facebook, I entered the password for the account (which was no longer active) and at that point, I turned OFF all of the integration settings (calendars, contacts, etc). THEN I could delete the account.
Hope that helps
