There are some medical transplants that do not cause intrinsic harm to a living person and we call them living donors. One could give a kidney, a portion of one's liver, a lobe of one's lung, or bone marrow to another person. Morally, I think this might fall under the category of self-sacrifice to help a sick child or other relative. I do not believe, however, we should be forced to do so or condemned because we do not. There is great reward in suffering, so I would think these types of donations would have to be well thought about and prayed about.
As a former health care worker (in a lab sometimes with pathology), there are very few things that can be donated after one's death. I have helped on several occasions with the harvest of such things. You can use skin, bones, and eyes. Corneas are removed shortly after death and may be kept up to 5 days before they cannot be used. You probably could put in this category teeth as well. This is done quite frequently and it seems Pope Pius XII deemed this would not be desecrating the body.
However, what we are dealing with today is heart harvesting. When a person dies, the organs die as well. Once they are dead, they can not be "restarted." It is impossible. This whole living organ donor issued is due entirely to heart harvesting. As I mentioned above, there was absolutely nothing stopping a person from donating organs after a patient was dead. But death meant absence of a heart beat and respiration. (that is still death). Many, many attempts were made to transplant hearts moments after patients died. They were all unsuccessful because once an organ is dead it cannot be restarted. In 1964, Dr. James Hardy transplanted a chimp heart into a human in Mississippi. The patient lived for just over an hour. In 1967 Dr. Christian Bernard got permission from a man who daughter had been severely injured in a car accident. He told the man that his daughter would forever be in a coma, but that with his permission they could let her "die" and transplant her heart into another man. To get around the laws of removing living organs from patients, Dr. Bernard injected her heart with potassium which sent into into shock. Then he cut it out of her. Mind you, her heart was not dead. If it were, it would not have been restarted. It was just shocked.
Following his limited success, Harvard established a committee to push-ahead with the concept of brain-death by using the idea of "irreversible coma." This unleashed a series of experiments in medicine throughout the 60s and into the 70s. Currently, brain death is the standard for determining death---and it has everything to do with the harvesting of organs. Morally, this is unacceptable because a living person, who is unconscious, would never give consent to have their vital organs removed. Ever. Thus, they manipulate the definition of death and the emotions of the families. The post conciliar Church went along with this atrocity by accepting the myth that brain death is actually death. Couple that with the concept of "irreversible coma" and we have people "coming back to life" rather frequently when they weren't even dead.
It's a whole big deception and the only people profiting are the doctors.....