There is no psychological name for the fear of elevators. However, an elevator is often a common trigger for those who suffer from claustrophobia and agoraphobia.
Claustrophobia is the fear of small enclosed spaces, such as elevators. Agoraphobia is the fear of being trapped in a situation that would be hard to escape, such as being stuck in an elevator. Both phobias can cause panic attacks and high levels of anxiety for the person dealing with them. These two phobias often go hand in hand, prompting many sufferers to take the stairs each time they are in a building with multiple floors.