In this teshuva from 1967 (even earlier than those from Rav Moshe Feinstein), the Tzitz Eliezer begins by stating that removing a heart that is still beating in order to transplant it into someone else is tantamount to murder, based on the Gemara in Shabbat 151b that rules that any action which hastens death is considered to be murder.
However, Rav Waldenberg then raises the possibility that the donor is considered to be a treifah, since he is unlikely to live, and perhaps this avenue needs to be explored in terms of potential room to permit taking the heart from someone who is not yet clinically deceased. However, he rejects this position on the grounds that while a court cannot condemn someone as a murder for killing a treifah, that person nevertheless has the status of being a murderer. Furthermore, Rambam in Hilchot Yesodei HaTorah rules that one may not kill another Jew for the purpose of healing another, which would seem in modern times to apply to such a case.
However, the Minchat Chinuch comments on that Rambam that if non-Jews are threatening to kill a group of Jews unless they hand someone over, they are allowed to hand over someone who is a treifah in order to save the rest of the group. While this seems to be another avenue that could lead to permitting taking a live heart to save someone else, Rav Waldenberg rejects this for three reasons. First, as stated before and as pointed out by the Noda BiYehuda, we may not actively kill a treifah, and thus the inference of the Minchat Chinuch is incorrect. Second, it could be that only the person who is himself a treifah is allowed to hand himself over, but no one else is allowed to do so. Finally, whereas Rambam is discussing a case where the non-Jews specifically asked for someone to kill, in this case the doctor is not looking for this particular person per se, but rather for anyone who could serve as a donor, and thus the logic of “whose blood is redder” kicks in.
In terms of the recipient there is a problem as well, since the chances of survival with the new heart are not guaranteed, and thus there is the potential problem that removing his diseased heart and replacing it with someone else’s heart will result in the death of the recipient. Even though every surgery carries with it the potential of danger to the patient, such surgeries are permitted so long as the chances of death are minimal. However, since in the case of heart transplant surgery the chances are death are quite great, it is not permitted under the general heter that is given to surgery.
Rav Waldenberg concludes by speculating whether such a surgery, the likes of which was never envisioned in earlier times, should even fall under the rubric of ורפא ירפא, which is the source allowing physicians to practice their craft. [ed. Note – I am not sure what he means by this, since taken to its logical conclusion, this reasoning would forbid using any innovation in medicine. If anyone can shed light on this, I would be most appreciative.]