Revealing mistake: When Hawkeye exits the tub, you can see his shorts as he wraps the towel around himself.
Visible crew/equipment: When Klinger's in Colonel Potter's office wearing the fur coat and rubber reducing suit, just as Potter says, "This one is definitely in the running," we can see the shadow of the boom mic shaking up and down on the wall, at the left side of the screen.
Visible crew/equipment: As Klinger sits in the chair dressed in a winter coat, you can see the boom mic and light panel reflected in his shades.
Audio problem: As the Scrounger comes up to Hawkeye and BJ as they walk out after visiting Radar, you hear the horn honk, but the horn button never gets hit.
Answer: Even an Army field hospital would have SOME weapons for self-defense if the enemy attacked. What's meant is that it's not a combat outfit, and therefore wouldn't have enough weapons to actually attack anyone. (It's also mentioned a few times that the Officer of the Watch is supposed to carry a pistol at all times, but Hawkeye refuses to do it when he's in charge and Colonels Blake and Potter don't enforce the rule).
Captain Defenestrator