Can you exhume anything other than a body?
Solution 1:
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)
Solution 2:
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.
Solution 3:
In one sense, I think the question is the wrong way round. We tend to use "exhume" for bodies, because "digging up" sounds rather sordid. It is more that the word implies reverential removal from the earth, and so could be applied to treasure or other artefacts.
However, in most cases, it makes more sense to just say you are digging them up. We seem to have a problem if we tell a relative that we are going to dig up their dead relatives corpse, so we use other words to say that the process is being done with care and reverence.
Strictly, as others have pointed out, anything being removed from the earth could be exhumed.