Near the end when Hogan is sitting at a table drinking a beer (playing the part of Major Hochstetter), Lily comes over to sit with him. The drink in the glass is foamy and just over half filled. When the camera changes to a closer look. The beer glass is full and flat. Then back to the wide shot and the glass is half filled.
At the start of the introduction, Hogan gets a fresh beer with a foamy head on it. When he salutes Lily with the beer, there is no foam. When Lily sits down next to him, Hogan sets the beer down, but it is foamy again.
The terms 'Gestapo; and 'SS' are used interchangeably in referring to the guards. In reality, they were different organizations.
There are several scenes where a production light is seen reflecting off of Klink's monocle. The bulb can be clearly seen in some instances. Update: When Klink, Hogan, and Hochstetter are first discussing Lily Frankel in Klink's quarters, there is a cut to show Klink, and he says, "Hogan, please." His monocle shows the reflection of two stage lights.
When Hochstetter orders one of his guards to arrest LeBeau and bring him to the cooler immediately, the guard salutes, but stays standing in place. He should have started marching to the barracks to get LeBeau.