When Paul Newman is riding on the tractor, they are clearly driving down a dirt road. When they stop, he gets off and the tractor turns around. Behind the tractor is a plowed field, not a road.
When Professor Lindt is erasing the chalkboard of his calculations, two shots show him erasing only the right side. But in a subsequent shot, the entire chalkboard is erased.
In East Berlin there are several Volkswagen Käfer / Beetle on the street which is a west German car and definitely not would have been found in east Berlin. The car which they took from the airport to the hotel is a Mercedes Benz, a west German car as well.
A few snippets of dialogue in the scenes in the East German university clearly show that the extras are Americans.
The film is set in early September 1965; that year, the Tivoli gardens, where the restaurant scene in Copenhagen takes place, were open from May 1st to September 30th. During that period, the temperature would never be at freezing point in Osterfjord, Norway, where the opening scene takes place.
The cars that bring Armstrong to the hotel are Mercedes Benz. The taxi he later takes is a BMW. These were not the cars used by the government (that would be Zil. Volga, Wartburg) or as taxi's (would be Lada or Zhiguli). East German and Russian made cars.
Armstrong has a PhD from CalTech, yet he is often referred to as Mr. Armstrong rather than Dr. Armstrong as they refer to Dr. Lindt.
When Professor Armstrong is on the boat, the heater aboard ship is broken, and the close-up of the thermometer shows it is freezing - yet it spite of the fact that he and all the extras are wearing heavy coats, their breath yields no steam, which would have been profuse at that temperature.
The handwriting Professor Armstrong gives to the radio operator aboard ship and the note that he later writes to his fiancée is not the same - both handwriting samples clearly do not match.
When Gromek's motorbike is unearthed, the chrome is shiny, without any soil on it.
In the farmhouse scene when Professor Armstrong is fighting with Gromek, Gromek is choking Professor Armstrong but Armstrong shows no signs of it, i.e. bruising, in later scenes.
Or their shadow, anyway. On the road-level shot of Armstrong's taxi leaving the farm (Gromek's motorbike is visible on the left of the screen), just at the very bottom of the image can be seen the shadow of the camera (4:3 television version only).
Reflected in the window of the farmhouse as Armstrong enters.
When Professor Michael Armstrong (Paul Newman) is writing the address on a piece of paper in the beginning of the movie he writes: Kobenavn. The actual Danish spelling is: København.
(at around 31 mins) During a scene in Berlin, a car parks on a zebra crossing.
Ms. Shepard states she is Dr. Armstrong's fiancee, but wears no engagement ring.