When the prisoners assemble for reveille while putting the dirt in the rafters, there is not enough time between the assembly time and the time that Franks comes in and finds the place clean for Bigelow to throw the dirt down the hole and clean the place up. He could not have cleaned that place up, as dirty as it was, in a minute or so.
When Sargeant Schultz and Sargeant Franks are talking in the barracks, the white coffee cup on the table changes positions slightly.
Carter indicates that the 'invisible ink' he has will become visible at 90 degrees Fahrenheit (32.5 degrees Celsius). Hogan follows by specifying that the temperature of the human body is 98.6 degrees Fahrenheit (37 degrees Celsius). While the internal temperature of the human body is generally 98.6 degrees (individuals may vary), the surface of the skin (where the clothing marked with invisible ink contacts Franks' body) is normally 89 to 91 degrees Fahrenheit (32 to 33 degrees Celsius). This would make the use of the invisible ink an iffy proposition at best, as it could not be counted on to oxidize (as that is the process that these kinds of invisible inks go through to become visible).
One scene, in color, shows Hogan looking out the land periscope; the next scene is black and white of historical footage showing old vehicles, not the ones that Hogan describes. The black and white scene is shown in two overlapping circles surrounded by black, trying to simulate looking through a periscope. It should be one full circle, as there is a single objective lens on the periscope.
When the prisoners straggle out of the barracks just before Franks's uniform starts falling apart, Newkirk and LeBeau can be seen wearing each other's caps in the wide shot, but in their close-ups they are wearing their own hats.
Update: This is not necessarily a goof. Newkirk and LeBeau swapping hats could just be part of the plot to discredit Franks (by being out of uniform). In any case, between the time they are last seen wearing each other's hats and the time they are correctly attired, there is a short scene where Klink says, "Sergeant Franks, this is shocking!" During that scene, Newkirk and LeBeau are not visible. The scene is long enough for them to have exchanged their hats again so they are correctly attired.
Update: This is not necessarily a goof. Newkirk and LeBeau swapping hats could just be part of the plot to discredit Franks (by being out of uniform). In any case, between the time they are last seen wearing each other's hats and the time they are correctly attired, there is a short scene where Klink says, "Sergeant Franks, this is shocking!" During that scene, Newkirk and LeBeau are not visible. The scene is long enough for them to have exchanged their hats again so they are correctly attired.
Bigelow is unable to leave the camp due to the escape tunnel being blocked yet he drives into the camp disguised as a Gestapo officer.
Update: Presumably, Kinchloe intercepted the call to the Gestapo and told the caller that they would not have someone out to pick up Franks until later that evening. 'Later that evening' would be after Max delivered the mail, during which Bigelow could have been smuggled out in the mail truck. After changing into the Gestapo outfit, Max's truck would be 'commandeered' to pick up Franks. The real plot hole is that no one notices that it is the mail truck which Bigelow commandeered (same license plate). Certainly, a truck that comes into camp every day delivering mail would be recognized instantly when it shows up as being commandeered to pick up Franks, and the oddity would be questioned.
Update: Actually, none of that supposition is necessary. When Kinchloe is coming up from the tunnel for the roll call, Hogan asks where Bigelow is. Kinch replies that he was "beyond the shaky part" or something similar, implying that he was on the other side of the collapsed tunnel. From there, he could use any one of the escape hatches from the tunnels to get out of the camp. He can then later re-enter the camp as the Gestapo officer.
Update: Presumably, Kinchloe intercepted the call to the Gestapo and told the caller that they would not have someone out to pick up Franks until later that evening. 'Later that evening' would be after Max delivered the mail, during which Bigelow could have been smuggled out in the mail truck. After changing into the Gestapo outfit, Max's truck would be 'commandeered' to pick up Franks. The real plot hole is that no one notices that it is the mail truck which Bigelow commandeered (same license plate). Certainly, a truck that comes into camp every day delivering mail would be recognized instantly when it shows up as being commandeered to pick up Franks, and the oddity would be questioned.
Update: Actually, none of that supposition is necessary. When Kinchloe is coming up from the tunnel for the roll call, Hogan asks where Bigelow is. Kinch replies that he was "beyond the shaky part" or something similar, implying that he was on the other side of the collapsed tunnel. From there, he could use any one of the escape hatches from the tunnels to get out of the camp. He can then later re-enter the camp as the Gestapo officer.
Just after Franks' first formation, he runs the POWs off to a work detail while Hogan walks toward Klink. The view then shifts to a two-shot of Hogan and Klink. As Hogan moves into position next to Klink, the shadow of the camera is visible on Hogan's right arm.
With the amount of dirt that fell into the tunnel, there would have been holes in the ground where the dirt had fallen, and the tunnel would have been found.
In the opening, Col. Hogan is using the periscope to watch a German convoy pass and identifies the vehicles as ammunition carriers. We next see stock footage showing not ammunition carriers, but Puma armored cars. Col. Hogan should have known the difference.
When Franks quotes the regulations on the picture height and bunk separation distance, he specifies that the picture should be 8 inches above eye level, and the bunks should be 27 inches apart (not 28). He should have specified those distances in centimeters. Germany was on the metric system at that time.