Factual error: In several episodes, the fence around the cooler is visible, and hanging on it, a sign saying "Eingang verboten" meaning "no entry." The correct German term would be "Zutritt verboten." In German, "Eingang" is the opening where you enter a building, not the act of entering one.
Factual error: While the barracks were reasonable replicas of the real POW barracks, there was one main difference. The original buildings were elevated to ensure any escape attempts could be stopped. In this series, the barracks are flush on the ground, which makes the tunnel access possible and easy.
Factual error: Baker and Kinchloe, the radio experts of the troupe, often use Morse code to communicate by radio. When they do, they hammer the Morse key in different intervals, but always just barely tap it. Morse code is made up of "short" and "long" beeps. To produce a "long" in Morse code, you have to hold the key down three times as long as you would for a "short". A tap would be a "short" - the beeps they are sending are spaced long and short, but that's not how Morse code works.
Character mistake: In several scenes all over the show (though not always!), characters use the command "Raus!" to send somebody away from somewhere. Raus, short for "heraus" literally means "out" in the sense of "out of a building", which would be "aus einem Gebäude heraus" in German. Used as a command, it always means "get out", never "get away from there" or "get lost", in other words, the addressee must be inside of somewhere to be ordered "Raus." One example would be Schultz sending the prisoners away from general Burkhalter's car in S05E13.
Character mistake: During the opening scene of every episode, Klink comes out of his office and salutes. Then it cuts to Sgt. Shultz who then salutes. The superior officer never salutes first, it's a sign of respect of rank. There is one exception for this in the American military where if the soldier is a Medal Of Honor recipient, the superior officer will usually salute first but is not required to.
Factual error: In many episodes, SS members of all ranks appear - the most notorious recurring character being Major Hochstetter. Curiously enough, Hochstetter couldn't have been a Major in the SS, simply because that rank didn't exist there. The SS used the SA rank system, not the Wehrmacht one. Hochstetter for example would have to be a Sturmbannführer. Colonel Feldkamp would have to be a Standartenführer. To avoid confusion: Hochstetter sometimes claims he is Gestapo, even when he's wearing an SS uniform (different mistake). However, he couldn't be a major there either - he'd have to be a Kriminalrat or Kriminaldirektor, because the Gestapo, which was in principle a civilian police organisation and wasn't half as closely integrated with the SS or the military as the series would have us believe, didn't use military ranks at all.
Other mistake: Throughout the show, they always show snow on the ground and ice on the windows. It is almost always freezing weather. They regularly use a periscope viewer that is placed it a 55 gallon drum containing water, yet the water never freezes or shows any kind of icing up.
Continuity mistake: Throughout the whole series, the guys access the tunnels through a trapdoor built into the bunk closest to what appears to be the east end of the barracks. What is interesting is the method used to make the trapdoor open and the bunk rise. sometimes, it's two taps, two slaps, or two pushes of a "key", a panel that's made to look like the side of the bunk.
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.
Plot hole: Hogan's men have that clever device in Klink's quarters, where they turn the oven from below to reveal the trap door. That however unplugs the oven from the chimney, you can even see the smoke pouring out of the oven in some shots. Often enough, they have the oven unplugged for half a minute or more. If an oven smoked into a room that long, the whole room would fill with smoke so badly it could actually lead to smoke poisoning. If nothing else, Klink would have the oven checked. Any device that can turn a heavy cast-iron stove flawlessly like that would by definition have to be quite massive and not disassemble easily. If Klink had the oven checked, that would certainly lead to the discovery of the turning mechanism.