When the gravely ill patient on whom Trapper operated is slipping away but suddenly stirs, Ugly John quickly puts on his stethoscope and grabs the patient's arm, listens for a few seconds, and says, "Blood pressure's still low...but it's better." However, Ugly John was listening to the patient's pulse, not his blood pressure, which would require a sphygmomanometer and would take a significantly longer time to ascertain than Ugly John allowed.
Hawkeye steps into the shower and the handles operate the opposite shower heads, but this is not possible. Since the shower tent would have been built using the minimum of materials, there would only have been one supply line feeding both showers, and the handles would operate inline valves in single lines of iron pipe. The only way to achieve the results you see on-screen is to remove the guts of the valves, re-plumb the tent so each shower is fed by its own line, and have each line manually opened and closed off-camera.
Several times while Trapper is operating ( removing shrapnel presumably near the heart) his hands move as if he were scrambling eggs , even looking away to have his forehead wiped.
When Radar plays the drums, he is playing a Ludwig Supraphonic. That drum was not introduced to the market until 1964, more than ten years after the war.
When Henry is on the phone with the hospital after hearing that his son is born, he calls his wife "Mildred". In other episodes his wife is repeatedly identified as Lorraine, Mildred is the name of Colonel Potter's spouse (Henry's replacement).