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.
Continuity mistake: When the woman is kissing Klink in town, she hands a note to the boy who puts the note inside the spare tire's hubcap. There was no tape to stick it anywhere, yet when LeBeau goes to get the note, it is stuck with tape to the center of the spare tire.