This question often comes up, and did so just yesterday on e-sangha. I thought I’d repeat the answer I suggested to someone else a month or two ago, where the animal involved was a cat:
I do feel for you, as a cat lover myself. Many Western Buddhists have been through this. I can only offer you my opinion, based on discussions with quite a few other people.
1) You have already given your cat a far better, longer life than nature would.
2) The ulitmate critereon has to be compassion.
3) It’s obviously got to the stage where all that awaits your cat is more suffering.
4) *If* there is something “bad” about the euthanasia, take the consequences on yourself (the “bad karma”, if there is any, and the heartache) gladly out of love for your cat.
I have known people who have wished that they could be given the same release.
Like I say, only my opinion, but one that is widely shared.
Obviously, if you also know any prayers for the dead, they may help a bit, especially with your own feeling.
Best wishes to you both
It really gets up my nose when people start to put rules and speculations about the way karma works (suggesting, for instance, that you should leave the animal to suffer so that it can “complete working out its karma”) above compassion.