Stupidity: A prisoner, Jack Williams, rats out Hogan and his team to the Germans, so they fake his death and smuggle him back to London for court-martial. What a stupid thing to do. They should just shoot him and be done with it. First, what if he escaped from captivity on the way through occupied Europe? Second, he has done nothing wrong. Being a grass isn't a crime. Third, Hogan and his men are the war criminals - they often engage in combat activities out of uniform. Williams is going to be acquitted.
Plot hole: In several occasions throughout the story, e.g. S4E18, Hogan attends parties at Stalag 13 wearing an immaculate US dress uniform. Where did he get that? Dress uniforms are not part of the usual kit a bomber crew brings along with them on their missions.
Factual error: When the SS guard is calling Sergeant Shultz after seeing the burning fuse, he addresses Shultz as "Sergeant", rather than the proper German word for Shultz's rank, "Feldwebel."
Plot hole: Hogan has his men play romantic music over the radio. Earlier they were afraid of the radio in the observation post being detected. Elsewhere in the series, they try to avoid sending too long to avoid being homed in on - which is correct. Now, if they were to play a whole record at once, wouldn't the homing devices pick up on the transmitter at the camp?
Factual error: About 8 minutes into show, Major Hochstetter asks Schultz how long Colonel Hogan (an American) has been running the camp, and Schultz says 3 years last November. This is impossible. The US didn't get into the war until December, 1941, He would have to have been captured November '42 or earlier but the war was over by November 1945.
Other mistake: Hogan is disarming a booby trap briefcase. The bomb expert tells him over the radio he has to decide to turn the handle clockwise or counterclockwise. The expert doesn't know which way to turn it. Hogan turns the handle counterclockwise and disarms the booby trap, and the expert says counterclockwise. There's no way the expert would have known which way Hogan turned the handle. (00:21:50 - 00:22:26)
Continuity mistake: During his experiments to create explosives in the tunnels, Carter has a pretty extensive set of glassware on his workbench. Necessarily he would have to replace (most of) that after each explosion - not to mention various light bulbs, furniture and other non-blast-proof stuff by the way. A lot of that glassware is specialty equipment, it would not be easy to come by even one set of in peacetime for a free civilian. In wartime, for an allied prisoner (even with the heroes' connections) it should be nigh impossible, and totally impossible to have an inexhaustible supply of the stuff. So we can either assume a giant plot hole, or treat it (as I did) as a big continuity mistake.
Continuity mistake: LeBeau meets a female underground contact in a movie theater. They have to kiss to avoid detection by the Gestapo. When LeBeau leaves the theater there are no lip stick kisses on his right cheek, but when he returns to camp he has some. (00:18:00 - 00:19:00)
Continuity mistake: Toward the end of the episode, Hogan turns Shultz around, grabs one of the grenades Shultz is carrying, and hands it off. The one grabbed was closest to Shultz's left side. A few moments later, Shultz turns his back briefly to the camera; there is now a space between the grenades, as if Hogan grabbed the middle one.
Factual error: Several times in the series, maps purporting to show Russia have the title "Union of Soviet Socialist Republics" (USSR). This name existed only after the war ended, before that, they were independent countries until they were annexed by the then Soviets.
Character mistake: When Major Hagel is going down the list of his knowledge of Hogan's Heroes, he mentions that their code name is Goldilocks. Throughout the series, the code was also Mama Bear, but the most used code name was Papa Bear.