In this teshuva from 1968, Rav Moshe Feinstein offers an in-depth analysis of the issues involved with heart transplants. Bear in mind, as mentioned in the last teshuva, that this was written in the early days of heart transplants, when success was not assured.
Rav Moshe begins by flatly prohibiting heart transplants on the grounds that it involves the murder of two people - the donor and the recipient. The donor is deemed a murder victim since his heart was still beating when it was removed from him, and the recipient is deemed a victim since he is sure to die soon as a result of the transplant and he is condemned to a brief life of pain and suffering in the interim.
The bulk of the teshuva is divided into four sections, in which Rav Moshe makes the following main points:
1) It is forbidden to shorted someone's life by even the smallest amount, and thus so long as someone's heart is beating, removing it in order to give it to someone else constitutes killing that individual.
2) We are not experts in determining the exact time of death, and even once breathing has stopped it is still possible for someone to be alive.
3) Giving someone a heart transplant does not qualify as healing the recipient per se, but rather as prolonging his life. Since that life is likely to be one of suffering, it is forbidden to inflict such a situation on another person. [ed note - does anyone know Rav Moshe's opinion about hooking someone up to a life support machine? Wouldn't that seem to be the same thing?]
4) Rav Moshe's final point discusses the limits of the requirement of לא תעמד על דם רעך - that we are not required to actually cut off a limb in order to save another individual. Certainly in this case, where we are asking someone to give up a major organ for the purpose of possibly saving someone else for a small amount of time there is no requirement to do so.
[My hope is to continue to find teshuvot on this topic that were written as the medical science continued to advance. Stay tuned.]