As a disciple of the Supreme Buddha, what would you do with an animal that is in severe pain? You may get to see someone or an animal is sick and suffering in pain. You may be able to give that person or animal a medicine that helps relieve the pain somewhat. But sometimes there’s so much pain that it goes beyond the reach of any morphine, any opium, any painkiller at all. At  such moments what would be your thoughts? Being well informed of the Dhamma, you know in the back of your mind that it is our responsibility for what we are. We were the ones who chose to be born here. It was because of our desires and our cravings that we took birth as human beings or animals for that matter. When you’re responsible for your birth, then you also have to be responsible for how you handle your aging, illness, and death. They all come as part of the same package. So how could you be of help to self and the other in such a situation?
My mother and I saw a cat today by the side of the road, which had probably been hit by a car. Its back legs were outstretched, it didn't walk, I guessed its lower back was broken. It miaowed to us.

We went to find the owner of the nearby house, I told him that I had the regret to inform him that etc., he said that it sounded like his cat and we went to find it. He talked to it (saying, "Oh my poor Fifi") picked it up (it miaowed again) and he told us that he'd take her to the vet, which seemed to me like a good idea. I was glad for the cat's sake that she now had her human companion/friend with her.

My mother told him that (in her opinion) the cat wll probably want to be euthanased (a.k.a. put to sleep or put to death) but I think that Buddhist doctrine denies that and says not to kill, even in that situation.

Here are some references for how to behave with someone who is terminally ill:


There are many other references at A Buddhist Guide to Death, Dying and Suffering (the author is or was a chaplain at a medical centre).
And What nurses need to know about Buddhist perspectives of end-of-life care and dying (registration required) might be helpful.
Also page 82 of this commentary on the Vinaya describes the words a bikkhu might say during the dying process to inspire a patient.


The answers to these questions might be helpful if dying is accompanied by pain as well:


Experiencing physical pain
Buddhism view on mercy