Right after the introduction and mini-credits, Hogan visits Klink in his office, where he is eating cookies and drinking coffee (or tea). When the scene starts (the view is from behind Klink), there are five cookies on the plate (Klink is eating one). When the view changes to behind Hogan, there are only four cookies on the plate.
After the introduction and mini-credits, Hogan is in Klink's office, talking about the prisoners from Barracks 3 being moved into Hogan's barracks. The view is from over Hogan's shoulder as he reaches for a cookie on the plate. Klink grabs the plate with his right hand, as he tells Hogan that he doesn't have to share his cookies, Klink passes the plate of cookies from his right hand to his LEFT hand in order to put the plate on the tray to his left (to get them out of Hogan's reach). As Klink is doing that, the view shifts to being from behind Klink, and Klink now has the cookies in his RIGHT hand, his right arm is in motion, and he puts the plate on the tray with his right hand.
When the Heroes are dressed as women and are climbing into the truck, it is supposed to be at night. In a wide shot, you can clearly see light from the sun shining between buildings. Update: Normally, this should have been shot with a day-for-night filter. In post-production, they may have tried to dim it down some, but there is only so much that can be done in post. The segment where the gang distracts the guard with the burning fuse has the same issue.
When the prisoners light a fuse to distract the guard in front of the girls' barracks, the slow-burning fuse is being dragged across the ground to make it appear to be burning very fast.
In the credits, one of the women is listed as 'Ginger Flintrin' but in the show, she introduces herself as 'Ginger Wilson'.
When the girls are telling Hogan about their trip to Stalag 13, Kathy (Jean Hale) says "a few minutes after we started...", but her lips don't match the dialog.
When Hogan asks the women where they were captured, one of them says they were performing for an infantry unit when it was attacked and they were captured. Based on what they tell him, Hogan states they must have been 80 miles to the EAST of Stalag 13 (i.e., farther into Germany). This would mean they had been entertaining American soldiers far behind the German lines.
When the ladies are offered a chance of escape, one of them says,"Anything is better than being booked in a German POW camp for three years." She would have had to have known when the war would end in order to say this.
On a couple of occasions, Carter, who is not married, is clearly wearing a wedding ring.
When Colonel Hogan, Carter and Kinch are talking by the open bunk bed secret entrance, no one is watching the door for Schultz or Colonel Klink. They risk getting caught and exposing their operations by not having anyone watching the door.