Buddhists are not unanimous in their view of euthanasia, and the teachings of the Buddha don't explicitly deal with it. Most Buddhists are against involuntary euthanasia. Their position on voluntary euthanasia is less clear.
States of mind
The most common position is that voluntary euthanasia is wrong, because it demonstrates that one's mind is in a bad state and that one has allowed physical suffering to cause mental suffering. Meditation and the proper use of pain killing drugs should enable a person to attain a state where they are not in mental pain, and so no longer contemplate euthanasia or suicide.
Buddhists might also argue that helping to end someone's life is likely to put the helper into a bad mental state, and this too should be avoided.
Avoiding harm
Buddhism places great stress on non-harm, and on avoiding the ending of life. The reference is to life - any life - so the intentional ending of life seems against Buddhist teaching and voluntary euthanasia should be forbidden. Certain codes of Buddhist monastic law explicitly forbid it. Lay-people do not have a code of Buddhist law, so the strongest that can be said of a lay person who takes part in euthanasia is that they have made an error of judgement.
Karma
Buddhists regard death as a transition. The deceased person will be reborn to a new life, whose quality will be the result of their karma. This produces two problems. We don't know what the next life is going to be like. If the next life is going to be even worse than the life that the sick person is presently enduring it would clearly be wrong on a utilitarian basis to permit euthanasia, as that shortens the present bad state of affairs in favour of an even worse one. The second problem is that shortening life interferes with the working out of karma, and alters the karmic balance resulting from the shortened life.
Euthanasia as suicide
Another difficulty comes if we look at voluntary euthanasia as a form of suicide. The Buddha himself showed tolerance of suicide by monks in two cases. The Japanese Buddhist tradition includes many stories of suicide by monks, and suicide was used as a political weapon by Buddhist monks during the Vietnam War. But these were monks, and that makes a difference. In Buddhism, the way life ends has a profound impact on the way the new life will begin.
So a person's state of mind at the time of death is important - their thoughts should be selfless and enlightened, free of anger, hate or fear. This suggests that suicide (and so euthanasia) is only approved for people who have achieved enlightenment and that the rest of us should avoid it.