Assuming the result of enlightenment is attainment of Nirvana, and assuming Dalai Lama is enlightened, why does he keep on getting reborn?
Could it mean that Nirvana means something else, for example a state of egolessness while still a person on planet Earth?
This is a fundamental premise of Mayayana, that the goal is not to be extinguished and leave samsara like an Arhat, but to stick around until everyone is enlightened.
Here is the zen formulation of the Bodhisattva Vow (couldn't find the Tibetan version)
Beings are numberless; I vow to save them. Delusions are inexhaustible; I vow to end them. Dharma gates are boundless; I vow to enter them. Buddha's Way is unsurpassable; I vow to become it.
As for where one is upon enlightenment, there are more lots of formulations, I don't have a good idea on how to summarize them.
If I can add just a little more detail - the Dalai Lama is an emanation of Avalokiteśvara, the Bodhisattva of compassion. As @MatthewMartin said - Avalokiteśvara will continue to come into the world as long as there are unenlightened beings to help.
Interestingly Avalokiteśvara also eminates as the Karmapa and other Tibetan lamas so presumable can co-exist.
Mahayanas say that if you ask your teacher to come back, he can choose to come back out of compassion, he doesnt have to, he does to help sentient beings.
