Continuity mistake: While Klink is talking to Hogan and the Major, the chair next to the door is almost centered under a picture frame. When the Major is arrested the chair is moved to the left. When Klink sits down, the chair is centered again. (00:03:50)
Character mistake: In this episode, Hogan suggests a helicopter as means for escape from Stalag 13. As a matter of fact, the allies had no helicopters operational before April 1944. The famous R-4 made its first flight only in January 1942. So unless Hogan planned to steal one of the Luftwaffe's 20 FW Drache (a maximum of ten or so existing at any given time) or a whole bunch of the single-seat Flettner Kolibri (24 total built) he was out of luck.
Revealing mistake: LeBeau hands Schultz the pan straight from the hot plate, but Schultz puts at least three fingers around it while eating without any apparent issues. Note that LeBeau earlier said the Crepe Suzette were "not quite ready yet" - or in other words, still cooking. (00:01:40)
Continuity mistake: When Klink does a walk through in the barracks, he turns to face Hogan in front of Schultz. As he does, the end of his riding crop pushes against Schultz's lower chest and sticks. When the camera cuts the crop is free and well clear of Schultz.