Ghosts, Spirits, Apparition, whatever you wish to call it, is a common belief among and throughout humanity. This, in a way, is evidence of their existence.
It is believed to be some form of existence or life after death.
If such an existence is true, than they WOULD have to use energy of some sort.
What happens when energy is used? It is put into something else, and is no longer part of what used it. Put simply, 'it goes away'.
Now, follow along with me here...how do you restore or recreate energy?Sustenance, inhilation, and absolution. In other words, some form of matter or energy has to be taken in.
So YES, in a way, a ghost WOULD have to 'eat'.
Now, whether it can eat solid food, no one can really say.