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.
This is set just after the end of WW2. In the early scene where Lazlo is leaving on a bus, the jet stream from a jet airliner.
Jet aircraft did not become common until the late 1950s.
In Philadelphia, the sun only reaches 74° at summer solstice, yet the prominent skylight window is shown to project the sun vertically down to the altar. This effect is also earlier demonstrated by Adrien Brody on a model by means of a flashlight. The demonstration would not work, since a flashlight is not a parallel beam of light (like the sunlight or laser light).
Harry Lee Van Buren Jr. is seen driving down the road during the 1947 scene in what appears to be a porsche sports car. The U.S. did not start importing porsches until 1950. (This is the same scene with the jet contrail in the sky.)
Towards the end of the movie, László Tóth is told to have been born in 1911 and to have survived the Buchenwald concentration camp 40 years later in 1951. Buchenwald concentration camp was liberated and ceased operations in 1945.
Tóth is a highly unusual, if not impossible, last name for a Hungarian Jew to have, since it's common to a Slavic minority of Croatian descent of the region. But he's not supposed to be a citizen of present-day Hungary, but of the former Austria-Hungarian Empire, which joined the Adriatic coast.
At the Van Buren construction site they are using Tatra trucks, which is a Czech vehicle manufacturer - while there is no evidence pro and contra yet hard to believe that such trucks are imported by US firms, especially during the cold war.
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.
When Tóth and Van Buren are exiting the quarry the following morning, a modern solar panel can be seen affixed to the wall, to the right of the entrance.
The municipal transit bus shown during the opening credits is GM New Look bus, which was introduced in 1959. The story takes place in 1957.
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.
László states that he was arrested and sent to Buchenwald while living in Budapest, which wouldn't have happened in real life: this particular concentration camp was located in Germany, and mainly the German Jews were sent there, while the Hungarian Jews were transferred to Auschwitz in Poland.
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.