In the beginning scene, a prisoner (played by Roy Goldman) is the sentry on watch at the door for any approaching German guards. Suddenly, Sgt. Schultz barges in through the door and the sentry at the door has disappeared.
General Strommberger, in charge of a Panzer division, asks the prisoners to clear the roads of snow for his tanks. Tanks are tracked vehicles designed to travel through deep snow, mud and even moderately large trees. Shoveling the snow for tanks would be a waste of time.
When Hogan reaches for the icicles hanging from the front porch roof of the club, the icicles swing as if they are hanging by a hook or string - if they were actually frozen to the roof, they should have been immobile, or broken off when Hogan reached for them, instead of swinging the way they did.
The prisoners refer to Rudolph the Red-Nosed Reindeer. Gene Autry did not record that song until June 27, 1949, long after the end of the war.
UPDATE: This is not a goof. Rudolph the Red-Nosed Reindeer was created in 1939, before America was even in the war, by Robert May, an advertising copywriter who conceived Rudolph for a children's book that department store Montgomery Ward gave away in a goodwill and promotional gesture, distributing 2.4 million copies in that first year alone. May's brother-in-law Johnny Marks later wrote the song that Autry recorded.
UPDATE: This is not a goof. Rudolph the Red-Nosed Reindeer was created in 1939, before America was even in the war, by Robert May, an advertising copywriter who conceived Rudolph for a children's book that department store Montgomery Ward gave away in a goodwill and promotional gesture, distributing 2.4 million copies in that first year alone. May's brother-in-law Johnny Marks later wrote the song that Autry recorded.
When Hogan and his men are outside in the snow and cold weather you don't see their breath when they are talking.
At the start of the denouement, Schultz leads the prisoners out of the barracks. As they exit, there are several deciduous trees visible over the top of the building, in the upper right of the frame. They are full of green leaves, and there is no snow on them. At a minimum, the trees should not have had any green leaves. The other possibility is that the storm was a freak snowstorm (blowing in before the trees had a chance to drop their leaves) - but in this case, the trees should have been covered in snow.
Production lights are reflecting in Klink's monocle in some scenes.
Corporal Deitrich did not have corporal stripes on his uniform.
When the gang (with a bag of explosives and a thermos bottle) is first loaded into a truck, a guard comes back and orders them out. The guard confiscates the bag and thermos, and throws the thermos into the back of the truck, followed by the bag of explosives. At the time, thermos bottles had a glass inner lining, which was prone to shattering if the bottle was handled roughly. Therefore, the guard would have known not to toss the bottle into the back of the truck (assuming he was telling the truth about that now being his lunch), as the coffee would have been full of glass shards.