Can you exhume anything other than a body?

Yes, exhume can be used for the unearthing of things other than bodies.

See this recent story about exhuming buried WW II planes.

There is also precedence for using the term for bringing up sunken treasure and shipwrecks. (But maybe, technically, the treasure needs to be buried in the sand of the ocean bottom in order to be "unearthed," not just lying out in the open.)

The UN points to the discovery and archeological exhumation of the Pandora off the coast of Australia. We learned the true ending of the Mutiny on the Bounty and what happened to the mutineers by studying the wreckage. (Source for sunken treasure/ship article)

Sometimes it is used for that, although usually in order to evoke similarities to exhuming bodies.

There are things that are buried that are not bodies, however: flags, religious documents, and the like.

In the metaphorical sense, exhumation may refer to the revisiting of old feelings, as in the Randy Travis song, "Diggin up bones." ("... Exhuming things that's better left alone. I'm resurrecting memories..." )

Yes. Neither the OED's literal definition, nor the figurative, specifies what it is that is to be dug out.