At the beginning of the movie, which is set in the late 1940s, the Statue of Liberty is seen with its post-1986 torch and flame.
In a 1950s scene in Pennsylvania USA, during the card-playing, money put on the table includes US one-dollar bills with bright green ink, indicating they are Federal Reserve Notes, first issued in 1963. One-dollar Silver Certificates, having blue and black ink on the front, are appropriate for the era.
In the movie, Leslie Woodrow uses a shipping container as a site office. Shipping containers weren't invented until 1956, but work on László's grand project starts way before that.
New look fishbowl GM bus is out of place with 1940's sedans.
Harry's car in 1948 is a Porsche 356, which wasn't introduced in the U.S. until 1950, and wasn't common until the mid-1950s.
In between Lazlo's arrival in New York and going to stay with his cousin Philadelphia, there is a shot following the bus he takes down a country highway. In the sky (upper-left), there is a jet contrail. There would be a very low likelihood of a jet contrail in the sky somewhere between NYC & Philadelphia in 1947, since there were only 60+ jets at that time operating in the United States by the Air Force.
Throughout the film, László is seen sketching, writing and leading with his right hand. At the end of the first part of the film, however, a tight shot depicts a left-handed person who is implied to be László writing a letter to Erzsébet.