I heard the Dome of the Rock was built on the site where the Holy of Holies was, and that also it is on the site where the Third Temple will descend from heaven to earth. So will the Dome of the Rock be obliterated when Moshiach comes?
According to our sources Islam will become obsolete because Islam was based on false prophecy as explained here.
Hence there will be no need for it as nobody will be Muslim. The only use would be as a museum, and there's no reason to think that will be the case, especially since this will be prime real estate near or on the Third Temple area.