Will it hurt your bearded dragon if you dye it with food coloring?

No. As long as it is just coloring and no additives, no it won't be a problem. In fact they actually have some special water coloring that can enhance the color of a bearded dragon.

There is a product made specifically for this reason so obviously others have considered the same thing. There are special types of coloring made for dragons specifically that also enhance their color. Although normal food coloring shouldn't hurt them I would recommend a product specifically made for the animal you are planning to use the product on.