What is a term for a (fantastical) creature that consumes feelings and emotions for nourishment?
(Google searches amusingly turn up only discussions about "emotional eating" in the sense of people that over-eat when they're depressed.)
An Emotion Eater is someone who draws psychic nourishment or power from another person's emotions, and will usually do what they can to instill those emotions in others.
They are very similar to Psychic Vampires, ankle biters, energy vampires and other fantasy named creatures.
HT to @FumbleFingers for emotovore, which actually produced emotivore, used in Doctor Who.
Try emotional vampire. You may want to read this entry. http://en.wikipedia.org/wiki/Psychic_vampire#Emotional_vampires
The best term for this is psychic vampire. This is sometimes abbreviated psy-vamp (or psi-vamp). Alternate terms for these entities include energy vampire, energy predator, energy parasite, and energivore, as well as psionic vampire, pranic vampire, and empathic vampire.
You mean kind of like an incubus or a succubus? (It's just a thought . . ..)
In contemporary philosophy, there is the Utility Monster. Put simply, he gets so much pleasure from the things that please him that everyone else suffering in the world but him being happy leads to a happier world overall. In this way, it operates as a challenge to Utilitarianism.