When an animal dies at a zoo, what do they do with the body?

Responsible zoos have the policy of doing a necropsy (autopsy) on every single animal that dies on site.
This includes a general health assessment as well as identificiation of the cause of death. Not only is there a physical examination, but samples of blood and tissues are sent off for routine testing, as well as testing that may be associated with the cause of death.

Where possible, the necropsy is also used as a teaching tool - vets and vet students will use this opportunity to learn more about that type of animal. For many exotic animals in zoos there may be limited information about their physiology.

For many species, especially those who are endangered, there may also be a collection of sperm or eggs from the animal, for storage or study. This can be done even after the animal has died, in the process of "gamete rescue". Quite a few North American zoos now have this capability, and if they don't they are able to courier the testes/ovaries to a facility that can perform the rescue.
DNA samples (usually hair or a small section of skin) are often taken from the animal as well, to be used in research or frozen away for possible future needs.

Finally, the animal will be disposed of as appropriate for the species. Generally cremation would happen, but some animals would have their skeletons or hides preserved, or in some rare cases they might even be stuffed - for teaching tool purposes. This would be done by the zoo or an associated museum or similar organisation. Extremely large species (like elephants) are usually buried somewhere on the zoo site, space permitting.