Continuity mistake: Newkirk comes out of Klink's office with a code book, and passes it to Le Beau, who immediately holds it up and opens it. In the close up, Carter's hands bring up a book with a camera in it, and Le Beau brings the book out from his back and opens it again.
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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.
Factual error: In this episode, the HofBrau (which should correctly be written "Hofbräu") is displaying a big red neon sign above its door. From 1939 on, air raid regulations throughout Germany strictly prohibited any unnecessary display of light at night. Any light visible from more than 500m away was considered a breach of air raid regulations.
Factual error: In the restaurant in Hammelburg, where Schultz discovers Hogan and Newkirk, there's an advertisement for "Brauerei der Jager, Stadt Wien." (Wien = Vienna) Vienna and Hammelburg are more than 500km apart, and if Hammelburg were near Düsseldorf, where the series puts it, it would be more like 700km. That's a bit far away for a brewery to advertise in the pre-globalisation era.
Continuity mistake: Hogan dumps a bucket of water on Barnes and Davis each, to make it look like they were the ones who hid in the tanker. They get big wet patches on their uniforms from it, but aren't completely wet through and through - which is to be expected, after all it's just a small bucket of water each. After the next cut however, they are completely soaked like they went swimming.