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.
Factual error: The MP describes the antique vase Burns is supposed to have bought as "An 800-year-old seladon vase of the Ko-Yu dynasty". Later, Burns packs up a white vase to send to his wife. Seladon is by definition green, and that kind of color glazing definitely wasn't around in the 13th century. What Burns packs up looks most like early to middle Quing period - or rather a contemporary ripoff.
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.'
Revealing mistake: When Radar carries the field phone from Col. Potter's desk over to Hawkeye who is squatting in the corner, the phone doesn't have a phone line attached. It is an EE-8 Military Field Phone, which is a wired device, not a radio - even if Radar elsewhere says he has somebody "on the radio" when in fact he is using an EE-8.
Continuity mistake: As Hawkeye climbs under the table to hide there are three small stacks of white bandage boxes on the table. When the shelling starts, all the boxes in two of the stacks fall off the table, leaving one stack. When the shot changes, there are suddenly two stacks of boxes on the table.
Revealing mistake: In the shot where the horse kicks in the door, there's actually two shots - a kick, a quick cut, another kick. If you look closely, you can see how the hoof of one of the horse-leg props breaks off during the first kick. The second kick is performed by one leg only. If you look really closely - and probably use the slow-motion function - you can even make out the hand of the one wielding the leg through the hole made in the first kick.
Factual error: In this episode, Henry is seen handing Radar the keys to his jeep. While this probably benefited the understanding of the audience, it is historically incorrect. Jeeps assigned to a combat zone were outfitted with an ignition switch, not an ignition lock, for the simple reason that in an emergency the vehicle had to be useable by anyone.