I personally believe that Euthanasia is wrong and shouldn't be encouraged for killing is killing. We are not at the level of a Boddhisatva to transform the motivation into compassion hence our motivation may not be as pure. Only when it is, that the action is considered a virtuous action.
What if their karma turns out to be worse than it is now after they take rebirth in the next life? Since we are not at the level, we do not have the clairvoyance to see whether our actions are correct or not.
While they are still alive and although suffering, we can always do as many pujas as we can for them for a better rebirth, for their mind to be more at ease, purifying their negative karma and plant the seed of Dharma.
Found a teaching on this:
One must do everything within one’s capacity to prevent such a compromising decision, to ascertain that there is no alternative treatment or other method at all possible, and even then, the decision is not an easy one. Your decision depends on many factors. A sentient being’s consciousness does not stop after this life ends. If you were to kill a suffering animal in this life, but it’s karma at that stage was such that it were to be reborn into a lower realm, the pain that it is currently experiencing in this life (and that you are jeopardising your own karma for) is no different. In fact, such a suffering animal would be much better off for even just two minutes to remain in the pain it is experiencing in this life. Again, we can see it makes for a very difficult decision!
In the Mahayana teachings, killing, stealing, sexual misconduct, and telling lies are permitted in tantra, but to do these actions one needs strong compassion for the other sentient being. When bodhisattvas engage in these actions they have to have a brave heart and totally give themselves up to be born in the hell realms, so as to totally accept the heavy negative karma, the incomparable suffering in the hell realms, from the act of killing. Also they have to have wisdom, seeing that by doing these actions there will be great benefit. So, one needs strong compassion for other beings and wisdom to know what is beneficial.