Insofar as it relates to MH370 the treaty sections Pondlife quoted would probably work quite smoothly if it were found by a government or someone working on behalf of the govt. The question arises as to what happens when James Cameron goes and finds it. (Inevitable, am I right?)
If it is located by a private party then the law would not be quite so clear. It would fall to courts and there would be legal claims falling under admiralty law, salvage law and those might even be superceded by laws governing criminal investigation at sea.
Generally under ocean salvage law the original owner continues to own it, but the salvaging party is entitled to a substantial award or prize. To complicate matters, there are two international conventions on salvage law, one from 1910 and a second one from 1989. Some countries recognize one of them and object to the other.
If Malaysian Airlines is no longer in business when it's found that will make things even more unclear. Probably the best analog would be the Titanic case. The company that recovered most of the artifacts sued to be declared owner. A US court declared jurisdiction and ruled that as compensation for the cost of recovery they had the right to display the items, but they do not own them.
After a series of court battles, an American company, RMS Titanic Inc (RMST), emerged as the owner of the salvage rights, allowing it to keep possession and put on touring display the 5,900 artefacts it has lifted from the ship during six dives.
From an article in The Guardian
So far no one owns them since White Star Line is long gone. The US and UK quickly hashed together a treaty to declare them a public memorial so the artifacts can't be sold off to private collectors.
An international agreement signed by Britain and the US designates the Titanic as an international memorial and seeks to protect it from being plundered or damaged by unauthorised dives.
With all these things at play there is simply no telling what would happen with it.