(at around 33 mins) When Eve Black is in the hospital sitting on the bed talking with the doctor and he mentions introducing her to Eve White, she is holding a cigarette in her left hand. Her personality changes to Eve White, and covers herself, but the cigarette is gone.
(at around 34 mins) Eve's husband Ralph is sitting outside her room talking to Dr. Luther about her different personalities. As he listens to Dr. Luther (for about 2 minutes) trying to explain Eve's problem, the positions of his hat, his hands, and which hand is holding the hat change whenever the camera angle changes.
In at least two scenes, including the opening with Eve being driven to the doctor's office, there are cars in the background newer than 1951 when the story is indicated to be taking place.
When Jane and Earl are in his car and she explains her diagnosis, his voice is dubbed in two lines.
(at around 56 mins) When Ralph slaps Eve, it's obvious his hand passes below her chin, and the sound of the slap is a few beats late.