Factual error: Burkhalter refers to Jesse Owens winning the gold medals in the Berlin Olympics, and that Hitler left the stadium every time Owens was to be presented a medal. Truth was, Hitler left the stadium when another black man won the day before, and didn't snub Owens at all. In fact, Hitler wasn't in attendance the days Owens won.
Character mistake: Schultz states "In Hammelburg, they call me die Glitterzehe", which is a literal (but wrong) translation of the term "twinkle toes." That term does not exist in German, and even if it did, it would have to be GlitZerzehe, because the German word for twinkling is 'glitzern', the word 'glittern' does not exist.
Revealing mistake: During the main part of the plot, the guys are putting dynamite on each truck in the convoy as they fuel up at Stalag 13. It is supposed to be dark, (hence the reason the boys wear black) but there are shadows as if it were day. It's impossible for the driver in the following truck and the guard in the tower to miss seeing the boys as they go back and forth to plant charges.
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Factual error: In the rec hall where Klink is weightlifting, there is an advertisement poster for the Grindelwald region on the wall, which is in Switzerland. The German propaganda poured a lot of money into promoting the Reich's own recreational regions through the "Kraft durch Freude" program. With the war going badly, they especially wouldn't want a "Look how beautiful Switzerland is" poster in plain view of their troops every day.