I am not sure exactly what fish are predators of Damsel Fish, but I did find a list of fish that Damsel Fish are not compatible with (if you are trying to figure out what fish to put in your tank with one):

Incompatible: Eels, Groupers, Lionfish & Scorpionfish, Seahorses & Pipefish and Sharks & Rays.

Compatible: Dwarf Angelfish, Large Angelfish, Anthias, Basslets, Blennies, Boxfish, Clownfish, Goatfish, Gobies, Hawkfish, Hogfish, Parrotfish, Pseudochromis, Puffers, Tangs & Surgeons and Wrasse.

In terms of what fish Damsel Fish eat, they don't. Instead, they survive mostly on Algae, Crustaceans and Plankton.
Source(s):
http://animal.discovery.com/guides/fish/marine/damselintro.html

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