Based on Deuteronomy 21:22-23, which states that if a criminal is put to death by hanging, “his body shall not remain all night hanging on the tree, but thou shalt surely bury him that same day,” the rabbis conclude that to mistreat or mutilate the body of a deceased (known in Hebrew as nivul hamet) is a violation of scriptural law.
Since it is argued that the body of a deceased is in a sense being mutilated when an organ is removed for translplant into the body of a living person, strong opposition to the procedure exists. Most authorities agree, however, that when a transplant is likely to save a life, such surgery is permitted. Transplant surgery that results in the saving of a life adds glory and honor to the dead (in Hebrew kavod hamet). Thus, the positive commandment of saving a life is the highest priority, even superceding the laws of the Sabbath. The talmudic principle that is applied by the advocates of transplant surgery is zeh ne'heneh vezeh lo chaser, “one party is helped and the other is not harmed.”
Aside from the question of mutilation of the dead, many in the Orthodox community object to organ transplant surgery on the ground that it is a violation of Jewish law requiring that all severed parts of a person be buried. The question of organ donation has been addressed by Rabbi Moshe Feinstein, who counters that when any organ form the body of a deceased is transplanted into a living person, the organ can no longer be considered an organ of the dead; it becomes part of a living body, and the law demanding the burial of all parts of a deceased does not apply. Rabbi Isaac Klein, a Conservative rabbi, points out that a transplanted organ will eventually be buried, thus satisfying the requirement.
Source: Kolatch, Alfred J. The Jewish Book of Why/The Second Jewish Book of Why. NY: Jonathan David Publishers, 1989.