It completely depends upon the country. Under the common law, whoever finds the dead elephant owns the tusks if no person owns the land. If someone owned the land, then the elephant belongs to that person. Its weird, and its British so....
Either case, its still illegal to import the tusks to America. I don't know about other countries.
A better question would be: how soon do the tusks need to be harvested after death to be valuable? Do they rot? Can someone pick them up off the ground 6 months later and make some nice stuff out of it?
I would have no problem with people scouring the plains for already dead elephants. However, I doubt they die in that great of a number so as to make it worth someone's time to go out and look for dead elephants.