Factual error: Throughout the series, the SS and the Gestapo are often used interchangeably, as if the two organisations were basically the same. The most notorious example is "Major" Hochstetter, who sometimes claims to be Gestapo, sometimes SS, most of the time wearing an SS uniform. This is historically incorrect. The SS was a paramilitary and military organisation, while the Gestapo was a secret police force and since 1939 part of the Reichssicherheitshauptamt. The Gestapo, being essentially a plain clothes police, never had any uniform of their own.
Factual error: In several episodes, German hand grenades, the famous "potato mashers" are seen. In (almost) all cases the Stielhandgranate 43, easily identified by its pull pin through the base of the head, was used. If the series is set in 42, the grenades should be the 24 or the 39 model, because the Stielhandgranate 43 - as the name suggests - was introduced in 1943 and didn't see widespread deployment until 1944. The earlier models didn't have a pull pin but used a pull cord that ran through the handle and was hidden by a screw cap at the end.
Factual error: Throughout the show, German officers talk about "Nazi" this, "Nazi" that, even in official capacity. In fact, the mere mention of the diminutive "Nazi" could get you in serious hot water for the disrespect and dissent it implied. Correct would be either "National Socialist" or some reference to the Reich: e.g. "Officer of the Third Reich" instead of "Nazi officer."
Factual error: Klink usually wears an EK1 (EK= Eisernes Kreuz = Iron Cross first/second class) chest cross with a WW1 EK1 repeat badge, but he neither wears an EK2 ribbon nor a WW1 EK2 repeat badge. This is not a legal combination, he either has to wear both or none at all. The EK1 and EK2 repeat badges were awarded to soldiers who were awarded an EK1 in WW1 and another in WW2. To be awarded an EK1, you had to have the EK2 already. The Legal combinations would be: EK2 ribbon in the button hole with or without EK1 chest cross, EK2 ribbon in the button hole with repeat badge with or without EK1 chest cross, again with or without repeat badge. One legal way of wearing it is seen in S5E3, "The Klink Commandos", where Hogan wears a black-and-white WW1 EK2 ribbon with repeat badge and an EK1 chest cross with repeat badge. It doesn't make any sense for him to wear that (separate mistake), but the way of wearing it is correct.
Factual error: During the show, many (not to say most) higher officers are seen wearing a Knight's Cross with Crossed Swords and Oak Leaves, the fourth highest award for military valor of the Third Reich. In fact, a total of 177 of this and higher-ranking medals were awarded during the entire war, most of them in '43 and after. The series is allegedly set mostly in '42. Historically correct, most Knight's crosses should be of base rank or with oak leaves only, as of these categories more than 8000 were presented.
Factual error: In several episodes, Hogan and his men are confronted with mobile missile launchers. These are depicted as a missile on a flatbed from which it is launched at the push of a button. While there existed mobile German missile launchers for the A4 missile better known as "V2", they consisted of a whole column of trucks, among them a transport truck for the missile with erector hydraulics, at least two tankers for the fuel, a control vehicle and several other trucks full of equipment, not counting the transport capability for a dozen or more operators that were necessary to launch them. Also the missiles weren't launched from the transport vehicles, but placed on mobile launch racks that were transported separately.
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.
Continuity mistake: Throughout the whole series, there are references to the building that houses Hogan and company. Sometimes it's Building 2, sometime it's Building 3. Also, there are times that Hogan's building has a direct view of the Commandant's building, and times when there is a building between them.
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.