What would motivate an enlightened person to continue living? It seems like once somebody attains enlightenment, other than teaching, there would be no reason to continue living.
No, the will to live does not cease after enlightenment.
Firstly, we have to define what it means to have a will-to-live. Arthur Schopenhauer, he been the first who coined this term, defines the will to live as a drive or conscious reason to live this life (not future life), the will to live this life not the will of becoming after death which the Arhat extinguished. So, someone who doesn't possess the will to live will stop eating or even breathing as Zeno did and depart from life, but it's not so with an enlightenment person they have a reason to maintain this life.
As you pointed out an enlightened person will maintain his/ her life to teach others or if they can't teach they will remain in the world to ease the suffering of others by helping them to avoid the picks of pain and pleasure which are the worst enemy to a man seeking liberation.
"Ceasing the will to live" sounds depressingly suicidal. That's not a good description.
Dhammapada 203-204 describes Nirvana as the highest bliss.
You could say that the craving to enjoy life and, the craving to do or become something in life ceases.
When you look at your remaining years of life, and think that "in the next years or decades, I can travel to many countries, spend more time with my family, enjoy different types of food, build a career, become successful, raise my children to be successful etc." - well, that kind of thinking and emotions cease after enlightenment and is replaced with the peaceful bliss of the moment.
Correct. Other than teaching, generally there would be no reason to continue living (although there could be other compassionate reasons, such as looking after parents).
Attaining enlightenment is like running out of gas while driving. The car does not stop immediately and the driver has no desire to hit the breaks to forcefully stop it. Just like how the car stops when the momentum runs out, enlightened beings expire when their life runs out naturally.
Does will cease? No, it is the opposite, the will is all that remains. As my teacher said, once you "see" the emptiness, you have no reason to live, no reason to die, all you have from that moment on is your will, made strong through years of practice.
I'm not sure what is meant by "the will to live", but whether one has that view (or any other view!) consciousness continues no matter what. If by "losing the will to live" you mean the state of mind right before someone commits suicide, it is important to understand that simply ending ones life will not extinguish the consciousness nor the karma that impels one to take further birth. Moreover, an enlightened being's mind is in no way comparable to the mind of someone right on the verge of suicide.
The Buddha or other enlightened beings are no more capable of ending their consciousness than any other being. Mind and consciousness continue after the break up of the body. Further, the Buddhas and Bodhisattvas work tirelessly for the benefit of sentient beings. This is why the Buddha taught the Dharma and continues to manifest in countless ways to benefit transmigrating beings.
After Nirvana, that is the real enjoyable life, a state of supreme knowledge. A connection with cosmic consciousness and a cheerful acceptance of people and situations comes once we transcend in our practice and experience the inner world which is beyond the experience of senses and emotions. True bliss is in the company of such evolved souls. That serves the to the purpose of life. It's an enjoyable journey.