Elephants can survive without tusks. While most elephants have tusks, female Asian elephants don't have any, and some male Asian elephants are born without tusks. Removing an elephant's tusk completely up to its skull can be extremely painful for the animal and can lead death.
The tusks are elongated incisors in the elephant's upper jaw and are connected to the animal's skull. The extraction of the tusk means taking a large chunk out of the animal's face and digging into the skull, which can cause profuse bleeding and death. Trimming the tusk doesn't kill an elephant but prevents it from digging for food and defending itself from predators until the tusk grows back on its own.