We didn't really hurt it at all. Losing on the road to a team with a 112 RPI isn't going to hurt too bad, other than the missed opportunity to raise the RPI.
In fact, Tennessee improved their RPI this week after beating Auburn (RPI 63) on the road and losing to us on the road.
Another thing to keep in mind is that the RPI on Selection Sunday is the only one that matters. Hypothetically, you could have an RPI of 80 one day, win a game and see your RPI drop to 85, and have it look like that win hurt you, but it could turn out after everything is said and done that win actually helped you. (IE, if the game hadn't been played, your RPI on Selection Sunday would actually be worse.) So questions like these are really hard to answer.