The level of the water in Gladys's bathtub during the scene where Marilyn visits her at the asylum.
At first, the photo of Norma Jean's father is facing right, but in the next scene when he talks, it's been reversed and he's facing left. Also the plaster cracks in the wall behind the photo switch from left side to right.
Marilyn learns from Eddy that Cass has died. Charlie Chaplin Jr. died in 1968, 6 years after Marilyn.
3D footage of a fully formed baby is supposedly erroneously shown in the movie more than once to depict the fetus in Marilyn Monroe's womb when she was only in the first trimester of her pregnancy, when the fetus wasn't fully developed. However, this is more than likely an artistic choice to depict how Norma Jean thought of her pregnancy, rather than a genuine error on the part of the filmmakers.
Marilyn greets the Secret Service agents at her door with: "You were expecting maybe Mother Teresa?" Mother Teresa had not gained international recognition in 1962. It's highly doubtful Marilyn would have known who she was.
When Norma Jean goes to her hotel suite after the premiere, hoping to meet her dad, there is an electronic beep when she turns the key.