Could the forum problems be due to server issues? There are times when the server seems sluggish or unresponsive to me (it wouldn't load for me earlier today, for example).
It could be that the errors come when the server's lagging just a bit, something times out, and the database doesn't get all the information it wants, so it throws an error. That the errors often seem to 'go away' with a browser refresh also seems to indicate that might be the case.
Of course, I'm not ruling out phpBB/mySQL database corruption, or some other oddity.
If the version of phpBB fukufics is using isn't the latest, would it help to update it? I'm only asking because with another forum (non-fanfic) I frequent, the admin was asked a similar question, and he didn't think it was worth it. (Not that I'm even sure it'd help with this problem, but... eh.)