The plunger level in the syringe changes between shots.
During the final scene, when the Turk returns the jeep back with an unconscious Radar, Radar's wool cap falls back off his head just as the Jeep comes to a stop but it is back in place when the camera angle shifts.
When Lt. Smith is in pre-op with Col. Blake, he has mud splashed on his entire face. In the very next scene when he is outside with Hawkeye, only the left side of his face has mud on it.
Radar implies Lt. Smith's rifle is a select fire M2 Carbine (which was developed in World War II, not Korea), but it lacks a selector switch, meaning it is an M1 Carbine.
Radar states that the weapon that Smith has is "one of those new guns that can shoot thirty corporals a minute." That means that the weapon that he supposedly has is a M-2 (I saw the earlier entry about this really being a M-1). But if this really was a M-2 carbine, then it was hardly new in 1951-1953. The M-2 was introduced at the end of World War II in 1944. So, by the start of the Korean War in 1950, the M-2 was already 6 years old (hardly new to the US military establishment).
As Hawkeye sits down inside the tent, the back of the tent is missing its wall, as it is not fully intact. Plus the soundstage lights can be seen in the upper right corner.
When Henry grabs the syringe from the Turk and turns his hand over with the syringe stuck in it, the plunger falls of its own accord. This does not happen.
The Turkish soldier reportedly has a severely injured shoulder. Yet when Henry has his back to the Turk, who is rising off of his gurney, Henry returns to him and sets him back down on the gurney by grabbing both of his shoulders. A doctor would hardly have grabbed such a damaged part of the body in order to lie him down. But obviously not so an actor.
When Hawkeye is making the recording to his father he says that the day started out with a batch of wounded coming into camp. He states that they brought the wounded off the bus and began triage on them. He then explains that triage is the act of prioritizing the wounded to get those with the most serious wounds into surgery first. But on several occasions Hawkeye has mentioned that his father has been a doctor his entire adult life, so he should already be fully aware of what triage is.
When Lt. Smith points a M1 Carbine at Maj. Burns, Radar informs Burns that it is the New Carbine that "Just Came out." The M1 and its variants were all introduced before the end of World War II.