Continuity mistake: Hawkeye mails a letter to his father which is delayed in its delivery, and when Hawkeye finally receives a reply from his father it's dated May 24, 1951, but in the previous episode "Radar's Report" the weekly report that Radar typed up was dated October 17 through October 22, 1951.
Revealing mistake: After many hours in the OR Hawkeye takes a seat in the wheelchair that Radar's pushing, and when they reach the Swamp there's a wooden ramp on the ground right outside the door, which is never normally there, deliberately placed for this shot so Radar can push the wheelchair into the Swamp.
Other mistake: When Hawkeye and Trapper are arguing with Henry about not sending the little boy to the orphanage, Henry says, "Look, you think I wanna send this little fella away? I got two kids of my own. I'm a former child myself," but at the end of season 1 Henry's wife delivered a boy, whom Henry was aching to be with, so Henry really has three kids.
Continuity mistake: When Lieutenant Willis wakes up in the Swamp and puts his shirt on he's wearing his dog tags, but when he tosses the papers and puts on the rest of his clothes the dog tags have suddenly vanished, then they reappear, vanish, and reappear again.
Character mistake: While talking to Condon, Hawkeye and Trapper relate the story of Dr. Charles Drew, the man responsible for the storage of blood and plasma, and how he died due to not being allowed in a "whites only" hospital after being involved in horrific car accident. The true story is that he was taken to a "poor whites" hospital. Despite being black, he was admitted, and though the doctors worked diligently, his injuries were too severe, and he died. His family sent a warm letter thanking the doctors for their efforts.