Continuity mistake: When Radar walks into Henry's office to tell him that there's a general there to see him, we get a good view of Henry's new desk and its beautiful ornate design, but later when Hawkeye and Trapper are carrying the desk and then it's lifted out by chopper it's a completely different desk - also note that this desk has casters.
Continuity mistake: The "KILROY" Hawkeye writes on the window as BJ looks out was in the center with his fingertip. Later, as Hawkeye looks out, the "KILROY" is now closer to the frame, and the lettering is different, a small capital A in front of it, and lines above and below it.
Plot hole: When the truck with the orphans arrives, Col. Potter and Nurse Cratty greet each other like one would an old friend. Being new to Korea, Col. Potter doesn't know Nurse Cratty - as becomes clear from his reaction to Hawkeye's explanation to BJ earlier (who doesn't know Nurse Cratty for the same reason, but that's not relevant for this mistake). Note that in other parts of the series Col. Potter is depicted mostly as somewhat old-fashioned and reserved towards strangers.
Plot hole: When Hawkeye and BJ buy Margaret a new ring, it takes a typo in the inscription to give them away - which makes both of them and Mr. Shin almost clairvoyant. Not only did they somehow guess Margaret's ring size, but Mr Shin - phew, Houdini and Uri Geller together don't hold a candle to this guy - actually somehow divined the exact type and font of the inscription in the original so perfectly that Margaret actually only spotted the difference from the missing 'n.'
Continuity mistake: When Cpl. Hendrix is examined on the operating table, Klinger and Hawkeye admire his tattoos. Klinger asks "do you really think the real Louise is built like that?" and Hawkeye replies, "If she is, I'm in love". Later, when Radar talks to Hendrix, Radar says, "Her face is still pretty, and her legs are the greatest." Hendrix asks, "What about the rest of her?" and Radar replies, "There isn't any." How could Hawkeye and Klinger admire Louise's build if the tattoo was that shot up?
Factual error: Nearing the end of the show, we are told that the Chinese have become involved in the conflict and have attacked with some 300,000 soldiers. This apparently refers to massive attacks by the Chinese starting on November 1, 1950. Although the numbers are correct, the US/UN forces were unaware of the numbers. As late as November 6, the Far East Command continued to insist that there were no more than 34,500 Chinese communist soldiers in country. http://www.history.army.mil/brochures/kw-chinter/chinter.htm.
Deliberate mistake: With the show having lasted longer than the war, the producers apparently decided to turn back the clock to the very beginning of the War (June 25, 1950). To do this, it seems they have tried to make things as confusing as possible, intentionally mixing up facts, even when this openly contradicts the facts as presented in earlier episodes. For example, both B.J. and Potter arrived in or about mid 1952, but now are in the O.R. when the Chinese launch their massive offensive on 1st Nov. 1950, almost two years earlier. Trapper and Blake, who were there for the Army Navy Game which would have occurred Nov. 25 1950, are not. Many of the supporting clips also are too early or late for the events being discussed. For example, Winston Churchill's speech must have been before August 28 when the first British troops started arriving, likely also before the Canadians in July 1950, but the first MASH Unit was not set up in Korea until September 1950. Truman's speech to University of Wyoming students is actually from May 9 1950, a month and a half before the beginning of the conflict and 5 months before the first MASH was established, but the clip about his "Whistle Stop Tour" must be from the 1948 campaign. MacArthur's speech, on the other hand, is in response to the ticker tape parade of April 22 1951 when he was replaced.
Other mistake: Wendel, afraid of being shipped home, is trying to escape from the MASH by hot-wiring a Jeep. Despite just having had an appendectomy, he is leaning forward and to the right from a sitting position, which would have to put pressure directly on the incision, yet when discovered by Hawkeye, he easily and without discomfort sits up straight. Aside from him still being at the MASH, we know the surgery is recent because, just a little later, he winces and complains after making a very small movement.