I've been having the same issue as most. Disc Unreadable error when the second cutscene is starting or whatever.
Removing the Hard drive and attempting to play that way still caused a disc read error
Firefight seems to work 2 out of 3 times or so.
My guesses are (1) a commonality in our xboxes that odst is bugging out on. (I have one from about 6 months after launch that was rrod'ed and repaired). If so should be patchable as far as I'd think. or (2) it's more demanding than the other games I play in some way, and in doing so it's exposing a hardware fault I didn't know was there.
I will say, that a few weeks ago when playing the Wet demo on 2 or 3 occasions I got errors "reading the disc" (while it clearly was a downloaded demo). I didn't think much of it, just assumed it was a problem with the demo and it didn't really matter to me so deleted it and moved on.