The soul of a Jew is pure G-dliness. The fact that a spark of G-dliness manifests itself in this physical world, in a physical body, is truly amazing. Every Jew is, by his/her very existence a beacon of G-dly light on this world. G-d forbid to extinguish this light.

This is in a case where the person is in a vegetative state, and, seemingly, cannot accomplish anything. However, most cases of euthanasia involve people who are terminally ill, yet can still function in a limited capacity. Do you have any idea how much a Jew can accomplish with one action, word or thought? Every Mitzvah which a person performs creates an eternal bond with G-d. With one thought a person can do Teshuvah, instantly transforming him/herself from a wicked to a righteous person! It would be cruel to deny a person these opportunities.

[See also "Can one remove a consenting terminally ill individual from life-support?"]