Throughout history, descriptions of dragons and where they live have varied. Some have wings and sharp claws; others have flippers or no legs at all. Some dragons are large enough to wrap around the earth, while others are the size of small birds. Their homes range from caves to lakes. Most dragons, however, share a few features: scales, a snake-like body, batlike wings, a tail, razor-sharp teeth and, of course, the ability to breathe fire.

Like most omnivores, a dragon would have sharp teeth for tearing meat and flat teeth for grinding plants. It might also use its flat teeth to grind rocks. 

When a dragon grinds large rocks into bite-sized pieces with its molars, the rocks leave powder on their teeth. And when the dragon releases its fire breath, the rock powder acts as an 'on' switch. Voila! Fire comes from the dragon's mouth.

There's very little record of what dragons like to eat. Some cultures think that dragons liked to eat humans, other cultures have no record of what dragons might eat. What do you think dragons like to eat? 

Source: http://science.howstuffworks.com/science-vs-myth/strange-creatures/dragon1.htm