JOEL - A person who participates in euthanasia—the deliberate, intentional putting to death of a person suffering from incurable conditions or diseases—violates the commandments of God(Church Handbook of Instructions).
There is a difference between allowing a terminally ill person to die of natural causes(eg. discontinuing a life-support system) and the initiating of action that causes someone's death(such as a lethal injection).
The application or denial of life-support systems must be decided reverently, usually by competent and responsible family members through prayer and the consultation of competent medical authorities. It is not wrong to ask the Lord, if it be his will, to shorten the physical suffering of a person whose afflictions are terminal and irreversible by discontinuing a life-support system.