In "Red", who shoots Joe as he leaves Dunning's house?

Cooper’s reaction indicates that it did not come from the CIA, and the only other agency there was the FBI. Since it was not likely to be Frank, Sarah, Victoria, etc., it points to the FBI.

The problem of course is that just because Cooper called for hold-fire, does not mean that someone else from the CIA couldn’t have done it. As you suggested, it could have been another CIA agent working under direct orders from the higher-ups, working for his own agenda, or even just a case of nerves.

Also, Dunning probably has some more guards and could have tripped some sort of alarm to send for reinforcements.

Everything about that scene (especially the slow-motion and muted sound) make it clear that the shot is supposed to be dramatic and mysterious, so we are not really meant to know for sure. Plus, don’t forget that the scene leads immediately to an action escape, so the viewer is probably not meant to dwell on the facts and that scene served as a sort of transition to their unlikely escape.

I think your point will become clearer by the following scene from the movie:

Cooper tries to negotiate Frank's surrender, but Frank tells Cooper about the vice-president's treachery, which shakes his faith. Joe sacrifices himself by taking Frank's place and pretending to give up. An unknown sniper kills Joe as he leaves the mansion, despite Cooper's order to hold fire.

Source- Wikipedia

I actually think that it might have been possibly Frank that shot Joe (from behind) being that yes they needed a distraction to get out of the house, two the fact that Joe was going to die soon from the cancer, and three Joe was already thought to be dead from the agent sent to the nursing home.

Another possibility is Ivan. After the group makes the woods, Victoria and Sarah exfil to the rear, and after some manuvering encounter Ivan. That indicates Ivan was in the same direction as Victoria prior to the shooting. I think it can't be Victoria, partially due to the lack of reaction from Sarah, but also because Victoria is using a suppressor and the rifle used to shoot Joe is not. It's probably still more likely there's another CIA agent or an FBI sleeper with kill on sight orders from Cynthia.