Factual error: After Radar asks Klinger when his home movies will be ready, Max answers, "Ready whenever you are, C.B." This is the specific punch line to a joke that relates to the actual filming of Cecil B. DeMille's 1956 movie The Ten Commandments, which wasn't released until after the Korean War.
Continuity mistake: When the Champ falls forward, he gets wine all over his sweater - you can see it when Potter and Hawkeye lift him. After the cut, his sweater is clean again. Considering the position of the stains relative to Hawk's hand, at least some stains would be visible.
Continuity mistake: At several points, in the second half of the episode, you can see the identification number (2A401) on the general's jeep that Hawkeye stole. As well as the big star, showing it to be a general's jeep. At the end of the episode, Col. Potter shows up for the first time, in the same jeep.
Continuity mistake: When Hawkeye gets into the chow line and Igor suggests the peas and carrots, Hawkeye asks about the beans. As the peas and carrots are being scooped to be served to Hawkeye, he grabs Igor's arm and peas and carrots are spilled into the beans. Peas are VERY visible in the beans. The next shot with the beans, the peas and carrots have been removed from the beans.
Other mistake: During the surgery, after Hawkeye and Margaret reach the 8063rd, Margaret is explaining the procedure. She says that they are using a new vascular clamp designed at the 4077th. It appears that this is the clamp designed and built three episodes later in episode 16 "Patent 4077".
Factual error: At the end, after BJ has received his Bronze Star, Colonel Potter dismisses the formation with "At Ease. Dismissed." According to Drill and Ceremonies as I learned them, this is impossible. The only legal command from the at-ease position is "Attention."