No competent orthopedist would have given FDR crutches so short that he would have to lean forward and use them to walk on all fours, as Ralph Bellamy does. (Besides, the paralysis would have kept him from moving his legs.) Crutches should be long enough so that the user can stand up straight, support his weight on them and propel himself forward with his shoulder muscles.
In a 1924 scene, someone buys a newspaper with a Thomas Jefferson nickel. This design wasn't minted until 1938.
In January 1922, the Roosevelts read news about the Teapot Dome. This scandal did not leak into the public press until the latter half of 1923.
At one point, Louis Howe mentions helium being used in dirigibles. Hydrogen was being used up until the late 1930s, well after the historic setting of the movie.
Inside a car on the way back from one of Eleanor's speeches in 1924, the background is a Manhattan street scene from the late 1940s.
When the train departs to carry the Roosevelt family home from Campobello, there's a shot of the departing locomotive accompanied by typical steam-locomotive-starting-up sounds which are not paced or synchronized with the actual operation of the steam engine.