Well, no, actually it doesn't...I'm puzzled by this. Recognition by UNESCO as a World Heritage Site doesn't in any way change the ownership of the site or the land.
Designation as a UN World Heritage Site basically means that no development of any kind can be attempted within 30 miles of the site itself.
I happen to know this because I own some property around Zion Canyon in southern Utah. The UN has been trying to designate Zion and Bryce canyons as World Heritage sites. In the meantime, the largest known untapped deposits of coal and natural gas on the planet happen to exist in the region (have we forgotten Escalante Nat'l Monument, created by executive order of Billy Jeff already?). While the status of Escalante and the UN's dalliance in the state are being hashed out in courts, property owners are being denied the right and opportunity to profit from their property.
It has gotten so bad that many of the towns in southern Utah have passed ordinances aimed at making it clear that the UN and it's minions are not welcome at all. There was even a town ordinance in Virgin, Utah which stated that all adult males should be armed, just in case the UN showed up.
I haven't been back for about three years now, so I don't know if that actually passed into law (but it would be neat, wouldn't it?).