Continuity mistake: Frank hits the pole supporting the center of the tent, which lets the tent collapse. However, in later episodes, the tents, especially hers, has a sturdy wood frame and doesn't need a center pole.
Continuity mistake: When Henry is put on the table, he rolls onto his back while requesting gas. After a cut to Hawkeye, he's on his stomach.
Henry, Please Come Home - S1-E9
Continuity mistake: The message Radar is presenting to Henry goes from chest height on Henry to face height between angles.
Henry, Please Come Home - S1-E9
Visible crew/equipment: After Reveille has blown, Radar comes in, and gets items thrown at him. As the door closes, there's a reflection of the studio lighting on the glass.
Character mistake: Hawkeye and Trapper make Tuttle an alumnus of "Berlin Polytechnic" and translate that (wrongly) as "Berlinishes Polytechnikum". Berlin's polytechnic university was named "Technische Hochschule Berlin" at the relevant time - and at no time offered a medical curriculum. To study medicine in Berlin, Tuttle would have to attend "Freie Universität Berlin" instead.
Suggested correction: And how are Hawkeye and Trapper suppose to know that? They are after all, making all this up. It is never meant to be factual.
It is intended to fool Margret and Frank, so it has to be assumed that they meant to pull it off. Hawkeye says "It can't be any place they'll check", which is why they select a German university in the first place. That shows it is indeed meant to be if not watertight, then at least convincing. To use an university you don't know the proper name of doesn't make sense under these circumstances.
Other mistake: While celebrating the ceasefire, a Jeep full of people including Father Mulcahy stops between the Swamp and the main building. There's a streamer hanging down either side of Mulcahy from above the top of frame. Unless there's a power line to the Swamp that never appears at any time, there's nothing for it to hang from.