Different episodes have treated this topic in different ways. But your question is based on the premise that holographic characters can exist outside the holodeck. They cannot. "What happens to food that a holographic character is holding, who then leaves the holodeck?" The person cannot leave in the first place. To answer your question, the character would disintegrate, and presumably, so would the food or any other holographic material they were holding. In The Big Goodbye, a holographic character walked outside the holodeck and slowly faded. In Ship in a Bottle, Picard threw a book outside and it immediately disappeared. Take your pick, although personally I think Bottle got it right. Imagine the poor guy in Goodbye seeing himself slowly disappear. Pretty cruel if you ask me.
If you are asking about food that the character has eaten, you would not want to think of the food as being digested, like it is for us. Remember that the people are not people, the food is not food, etc. Better to think of them as characters in a video game or other simulation. What happens when they eat food?
I would assume that any real material they were holding would fall to the floor, although that has never taken place in the show. Nor is it addressed if the character can even hold anything real. In Star Trek Voyager, the holographic doctor can manipulate real medical instruments, but it is left unclear if that was a recent development or if it was always available.
The episode Elementary Dear Data has a blooper. Moriarty draws a diagram on a piece of paper, which Data carries out of the holodeck. He should not have been able to do that, as the paper was part of the holographic program.
