Hogan throws the stolen map into the ceiling light and notices that it is starting to catch fire due to the smoke and switches off the light. Next day when he retrieves the map from the ceiling light it is undamaged.
When quoting the original estimate for the painting job (from the local company), General Burkhalter says it is 450 Marks and 12 Pennies. He should have said it is 450 Marks and 12 Pfennigs (the 'P' is essentially silent).
He does say "Pfennigs".
He does say "Pfennigs".
When Carter mistakenly hits Colonel Klink's hat off with his broom, the insignia is knocked off the band on Klink's hat. The actors continue the scene regardless.
After Carter and Newkirk return to camp Schultz discovers them still in disguise. Newkirk has a half-smoked cigarette in his hand. There is a cut to Hogan and when the camera returns to Newkirk the cigarette is much longer. Update: The change occurs After Hogan says, "That's the trouble with this war - everybody's a critic." Prior to that, Newkirk takes several puffs on the cigarette during his lines about 'the letter', which shortens the cigarette.
There is an attempt to cover furnishings during the painting process, albeit poor. That said, Hogan's paint crew remain wearing their unprotected uniforms even while painting overhead in Klink's office. Highly skilled professional painters would not even begin to handle open paint whilst clad in their street-clothes. However, this rank amateur paint crew miraculously manage it all without a single paint drop on their clothing. Update: The logic behind this makes no sense, as the paint crew is a bunch of prisoners-of-war, and would never be given any protective clothing to do the paint job.
Painting a congested office with personnel actively performing their day-to-day activities makes absolutely no sense. Staff and painters will be constantly in each other's path, the fumes from the paint would be nearly unbearable and, it is inevitable that staff will repeatedly contact wet paint.
In scenes in Klink's office, a production light is reflecting in Klink's monocle, to the extent that you can see the bulb and the light reflector.
When Carter and Newkirk leave Luftwaffe Headquarters after trying to take pictures of the inside (and failing), there is a cut to Hogan berating them about the picture or pictures they did get - of the outside. Hogan has the pictures in his hand. There is no way that Carter and Newkirk would have stayed in costume during the entire process of creating the pictures - first developing the negatives, running them through a 'fixer', rinsing the negatives with water, drying the negatives, exposing the print paper to the negative, developing the print, 'fixing' the print, rinsing the print, and, finally, drying the print. The process would have taken a considerable amount of time, during which Carter and Newkirk would have changed back into their regular clothes.