In theory, maybe. However, there would have to be many more dragons than there are in the world, and require millions of hours of munching and snacking. Even then, it is unlikely that the dragons would want to destroy the world, since they live in it, too. That would be kind of like blowing up your house to spite your ass. It would also require a level of cooperation among dragons that really isn't practical, given the prideful, solitary nature of the species. Some dragons do occasionally serve another dragon, and in very, very rare instances you might get several dragons to follow one very powerful one--i.e. Veles Fraise--but this is exceedingly rare.
Dragons reproduce infrequently, so there aren't many of them on the planet. You would have to have a LOT of dragons to bring forth the nompocalypse. The average lifespan of a dragon is 14,000 years. In that time, a female dragon may reproduce twice, at most. Some never do. At any given time, there are between 600 and 1000 dragons living, including ones in hibernation, living in seclusion in areas humans find uninhabitable (Antarctica), or occasionally traveling to other planes of existence. I don't want to go too much into it the dragon's reproductive process, as that will ruin part of the books, but it doesn't happen a lot and they manage to keep their population stable. Females can control their ovulation, so a dragoness will not get pregnant unless she wants to.
Furthermore, even though dragons can digest just about anything, it still takes time. Their bellies do fill up and they have to rest for a while to burn through it. Also, there are some things that a dragon simply will not consume. You'd have a hard time getting most dragons to eat garbage or the waste of other creatures. And most dragons don't enjoy eating rocks, asphalt and concrete. Also, dragons continue to grow as they eat. It is rare that a dragon can eat so much that it become obese, but not impossible. Eventually, the ohmnomageddon would get stumble to a halt as the dragons became too overweight to practically carry out their insane scheme. As prideful creatures, dragons would consider such a physical state horribly shameful. Finally, this also means that, at some point, the dragon would have to eat its own treasure horde...something no dragon would ever do. No, they wouldn't eat another dragon's hoard, either: they'd just take it back and add it to their own.
Simply put, if a group of dragons wanted to destroy the Earth, they'd resort to some method more practical than turning the whole planet into a Golden Corral buffet. But, that's a neat question. I hadn't ever really thought of that until you asked.