Factual error: There are a couple of references throughout the movie to Ralph Fiennes' character as an "American diplomat" "of the League of Nations". The United States was not a member of the League, so there were no "American diplomats", as such, working for the League.
Factual error: The lead character, Ralph Fiennes, goes to the racetrack in Shanghai in 1937, where an American flag is displayed. The flag has 50 stars, although the U.S. had only 48 states in 1937. (Alaska and Hawaii were not admitted to the Union until 1959.) The error is easy to spot because the rows of stars in the 50-star flag are staggered whereas the rows in the 48-star flag are even.