I know that elephant tusks keep growing (slowly – ~18cm a year) throughout their lives so I guess they would grow back, at least a bit. (And from reading a bit more they seem to have good healing abilities in their trunk.)
The problem is that elephants use their trunks to dig for water, to strip bark from trees (to eat the bark), and as a weapon in fights. If they didn’t have trunks for a period of time then I think they would be at a disadvantage.
I guess another problem would be risk of infection in the open wound after the tusk was cut off.