ANSWER: Organ donations are the removal of healthy tissues from one person and transplanting them into another who needs them. At the present time, the organs that can be given to others include the heart, kidneys, liver, intestines, lungs, pancreas, bones, skin and the corneas of the eyes. The majority of donations occur at the time of death (some people donate one of their kidneys while they are still alive).
Concerning the donation of an organ, Jesus gave us a sense of the worth of our body parts in his Sermon on the Mount. In his message he said that it would be better to have an eye or hand (which leads us to sin) cut off and thrown away that to have our whole body be destroyed (Matthew 5:29 - 30).
The apostle Paul offers us an important principle regarding using our bodies (which, of course, includes an organ) in our service to God. He states, "I exhort you therefore, brethren, by the mercies of God, to present your bodies a living sacrifice, holy and well pleasing to God, which is your spiritual service" (Romans 12:1, HBFV, see also 6:13).
Jesus' definition of love is also relevant to our discussion of organ donations. He stated, ". . . love one another, as I have loved you. No one has greater love than this: that one lay down his life for his friends" (John 15:12 - 13).
The donation of an organ so that others might continue living seems to be a good application of the principles taught above by both Jesus and Paul. Such an act would be the literal interpretation of offering our bodies as a sacrifice (Paul), out of love, so that others may live and not die (Jesus).
Some, who are against organ donations, state that we should leave this world with the same body we came with when born. This argument is a bit of a stretch and has not been thought through.
Forensic science knows that, within a year after death, the flesh of a person's body buried in the ground (and not in an air tight coffin) is almost completely decomposed with little remaining. It does not take long for all our organs to return to the dust from which they were made (Genesis 2:7, 3:19).
On the other hand, an argument could be made that forbidding the use of one's body parts (organs), upon death, to save the life of another would be rather selfish (especially since you will not be using them anymore).
The decision whether or not to donate organs upon death should not be based upon the myth that a person needs to be "intact" when they are resurrected from the dead. God has the power to resurrect a human no matter how they died or where their physical components ultimately ended up.
In the vision He gave to the prophet Ezekiel, the Eternal shows that he is fully capable of resurrecting humans even if all that is left of them is dry bones (see Ezekiel 37). Organ donations should be based on the truth and a person's prayers to our Creator asking him to reveal his will regarding this potentially lifesaving decision.