In some of the shots of Trapper in the minefield, the sky is a bright blue, but when the chopper plucks him and Kim from the minefield, there is fog and a cloudy sky.
At the end, when the group watches the truck containing Kim and his mother pull away off to the left, Margaret and Frank walk away from the group off to the right. The camera moves to the left to follow Trapper as he steps out into the road watching the truck. In the next closeup shot of the group, Margaret and Frank are standing with the group again.
When you first see the kid in the mine field and they tell him to stop and stay, he sits on a large rock. Later when the helicopter comes in to pick Trapper and Kim up, the rock he was sitting on is nowhere to be seen.
When Trapper first sees the letter from his wife, he says, "it's from Louise!" But when he reads the end of the letter, the wife's name is Maureen.
When Trapper, Hawkeye, Hot Lips, and Frank find Kim in the minefield, Loretta Swit puts her hands on top of a sign. Her fingernails can be seen to be an inch long. No nurse, especially a head nurse, would have fingernails that long.
When Trapper decides to adopt the injured Korean boy Kim he writes home to his wife to get her approval. Colonel Blake had previously told Sister Theresa that Kim would be all right in a couple of days and later changed that time line by a few more days. Considering it could take well over a month for a letter to be received back home there was no way Trapper could mail that letter and receive a response in 7-10 days. Major continuity issue.
In Korea the surname is first and the given name last. The mother refers to the boy as Kim. Kim is a surname in Korea shared with 20% of the population. While one of the MASH personnel might call him Kim, a Korean or certainly his mother would not.
When Trapper is writing a letter to his wife, he is only making wavy lines on the paper, as he never makes a capital letter, nor does he dot an "i" or cross a "t'.
When Henry is putting in his office, there is a model of a helicopter hanging behind him. The model is that of a Bell UH-1 Iroquois (Huey), which wouldn't take its first flight until 1956 (as the XH-40). There is no way that a model of the aircraft would have existed during the Korean War.
Hawkeye tells Trapper to stand still, that there is a mine 6 inches in any direction. Trapper's last movement was a half step to the right. Hawkeye must be reading the map wrong or Trapper would've tripped a mine before he made his last move.
Henry mentions that he has two kids, but in "Showtime", he learned that his son Andrew has been born and exclaims that he will be a father for the third time, so Henry actually has three children, not two. Although, one may chalk this up to the stresses of war plus the fact that he hasn't actually seen Andrew in person and hasn't yet adapted to having three kids.