When Hawkeye is calling certain personnel to the mess tent by name, he does not include Ho John, who shows up anyway.
The length of Hawkeye's cigarette changes from long to short and back again.
During his inspection of the camp, Henry is soaked on the front, top, and back of his head and shoulders. When he walks into the Swamp, the top and back of his head are dry.
When Margaret calls General Barker, she violates the chain of command within the 4077th. She would need the commanding officers approval to do so. Even though she is a Major, being in the nurse corps she has absolutely no command authority except where the nursing staff is concerned. In the 1950's , even the most junior second Lieutenant line officer would outrank her.
When Henry gives Leslie the fishing reel, he says it has a jeweled "escarpment". The more correct term would have been "escapement". An escarpment the slope of a plateau. An escapement is a latch release mechanism that you would find on a fishing reel.
When Hawkeye calls for a meeting on the P.A., most of them have arrived before their names were called. This would not be possible unless they knew ahead of time.
When "Henry" is checking the chimney in the Swamp, he flinches and closes his eyes in anticipation of being hit in the face by the soot.
When Henry presents Leslie with the "SatoTanaka " fishing reel, the box belongs to a Penn Beachmaster.
When Henry is presenting the fishing reel to Lt. Scorch, he doesn't notice that the box would be much heavier if the reel was in it.
In the closing scene, Father Mulcahy is speaking, saying "...it was after he returned to the garden he discovered that three of his disciples were missing." Presuming he's talking about Jesus and the garden of Gethsemane, he didn't return to find them missing, but to find them sleeping. While this is done for the comedic effect to set up Hawkeye's response, it's still not a mistake or misrepresentation a priest would make.
When Henry locks his desk after checking the stolen items, he doesn't actually lock the drawer, he just touches the lock with the key and jiggles it.