Organ transplants are less a problem in themselves than in two other questions. In the case of live transplants (where the donor is alive at the time of transplantation), the central question is whether the donor is allowed to endanger him or herself, and also to suffer the injury involved. It is generally prohibited to injure oneself, except for a valid reason (so that, for example, life-saving surgery is clearly ok, but self-mutilation is not only a bad idea, it is prohibited by Jewish law). This has to do with remembering that Judaism does not endorse the view that we are in complete ownership of our bodies; they were entrusted to us by God, with certain parameters around their use, and wanton self-injury violates those parameters.
That is why, years ago, R. Moses Feinstein ob"m was opposed to kidney donations. Since, at the time, the technique had not yet been perfected, and there was some danger to the donor (along with some uncertainty as to how well the transplant would take), he felt that it was taking on an unacceptable risk. With advancements in medicine, it has become more widely accepted. Lesser donations, such as blood, platelets, and bone marrow, are generally seen as meritorious, let alone acceptable.
When it comes to donation after death, another issue arises. Current medicine requires that the organs be removed from the donor body either before or within short minutes after the cessation of circulation, because the organs began to deteriorate so quickly after that. In the general culture, the assumption that brain death constitutes actual death solves that problem. While brain function has ceased and is irreversible, the body continues to pump blood, keeping the organs alive.
For Jews, the question is whether that constitutes death according to Jewish law. If not, to remove the heart or other organs of such a person would be tantamount to murdering them, since Jewish law is clear that hastening a person's death, no matter how certain or close that death is, constitutes murder. There is, however, a debate among rabbinic decisors as to whether the secularly articulated standards of brain death match halachic parameters. Many rabbis say yes, and support organ donation (there is an Halachic Organ Donor Society that promotes this view), but others disagree, and say that death only happens with the cessation of breathing (largely ruling out organ donation, not because of a problem with the donation itself, but because by the time breathing stops, the organs are no longer fit for transplantation).
May we be protected from these kinds of issues, both the need for the organs and the ability to give organs such as those, and be blessed with health for years to come.
Answered by: Rabbi Gidon Rothstein