However, humans will eat turtles, including snapping turtles.
Common snapping turtles have few predators when older, but eggs are subject to predation by crows, mink, skunks, foxes, and raccoons. As hatchlings and juveniles, most of the same predators will attack them as well asherons (mostly great blue herons), bitterns, hawks, owls, fishers, bullfrogs, large fish, and snakes.
As the other answers indicate, snapping turtle eggs and hatchlings have various predators.
I happened to see a show on discovery channel this week that mentioned that alligators jaws could easily handle a turtle shell, so I assume gators eat adult size turtles.
I found these comments on the internet:
In the northern sector of their range, many snapping turtles hibernate. Their habit of piling on top of each another to hibernate places the topmost turtles at risk of predation by otters.
Once they have reached a shell size of approximately 3 inches, snapping turtles have few natural predators other than humans.