What does Godzilla eat?

Interesting question. My guess is that he mostly feeds on any form of nuclear energy. Given his physiology, he does have the tools for being a predator as well. Taking that into consideration, he could potentially hunt down on large marine mammals and fish. Not like dolphin or grouper sizes (they are big though). I'm thinking more along the sizes of blue whales, humpbacks, finbacks, whale sharks, basking sharks, and any other bus-sized marine animal. But I feel nuclear energy is more efficient; it lasts longer and has more resources.

No amount of food would realistically be enough to sustain a creature of that size, let alone one as active as Godzilla, especially not one that fires an energy beam from its mouth. Fortunately for Godzilla, though, he doesn't need to eat, because his body is a living nuclear reactor, which gives him all the energy he needs (although he does seem to need to feed on nuclear power plants in some incarnations).

I do agree on feeding on nuclear energy. But as we saw with Godzilla vs Destroyah, in some cases this has an ultimate downside effect. 

maybe it is photosynthetic, like the plants, or superman (?)

The nuclear energy-as-food aspect is from the previous versions of Godzilla. Since this is a reboot, I don't think that will be the answer; after all, where would G find nuclear energy in a natural form? On the other hand, there wouldn't be enough organic matter to feed something that size...

Gareth Edwards might leave that as a mystery, with the human characters in the movie wondering the same thing: what does he consume?

Many of you seemed to have missed the point of the main question, save for LINKZILLA.

Realistically, Godzilla would never be able to sustain its own existence with anything less than several thousand gallons of pure rocket fuel fed to him continuously on a conveyer belt without end. Even then, Godzilla would be able to do little more than lie there and blink at us. Roaring, walking, using the thermonuclear breath. All of that would be entirely impossible for a being of his size and weight.

Realistically, Godzilla could never exist. Nor could any Kaiju above a certain size.

Just read this if you don't understand why: http://www.angelfire.com/ego/g_saga/kaijubiologyarticle.html