Did you know that a Rhino's horn and an elephant's tusk can be safely removed by trained zoologist without harming the animal? In fact they do it to elephants in captivity that destroy fences[0].
AFAIK, they can't remove the whole tusk. For one thing and for another, even if we ignore the morality of this, how do you ensure who removed the tusk?
[0] http://www.ncbi.nlm.nih.gov/pubmed/24614538