When the Vogon captain is reading his poetry, Ford's hairstyle changes noticeably throughout the scene; the result of shooting five months apart. (Of course, it could be that the poetry is so horrible that it made Ford's hair curl.)
When Zaphod Beeblebrox is at the controls of the Heart of Gold, snowflakes appear in his hair. There was supposed to be a scene in which Trillian creates a snowstorm while Zaphod is sunbathing. That scene was filmed, but was cut from broadcast. This is the only known deleted scene from the series.
When Marvin mentions that the ship has been stolen, and walks toward the camera to enter a doorway, David Learner's nose can be seen through the mesh of his mouth.
When Zaphod meets Arthur and Ford, the person operating his prosthetic arm is visible.