In the 'Dr. Heidegger's Experiment' segment of the film, while Vincent Price and Mari Blanchard are having a conversation in front of the portrait of Blanchard's character, the shadow of a boom microphone can clearly be seen moving back and forth above and between the shadows of Price and Blanchard.
After Gerald Pyncheon (Vincent Price) discovers the burial vault, he has the lid to the vault almost back in position, and after a quick cutaway, he's shown with the lid standing up once again about to place it.
Dr. Carl Heidegger is shown to be able to tell the mineral content of water just by looking at it under a microscope. In reality he would use chemical analysis and spectroscopy to determine such things.
As Alex and Sylvia finish their conversation while Carl is upstairs getting her wedding gown, they are both standing, and Alex is holding Sylvia by her upper arms. When Carl comes down the stairs with the gown, Sylvia stands up from sitting in the chair (she was standing just a minute ago) and Alex is holding a drink that wasn't there before.
Rappaccini's Daughter: When Giovanni reaches for the spider, the jet of smoke can be seen spraying from his palm,long before he actually touches the spider, which is where the smoke is supposed to come from as it burns from his touch.