Quaeritur: I was considering a career as a heart surgeon, but a friend told me that it is a mortal sin for a Catholic to have a heart transplant because in taking someone's heart, you are killing him. Is this true?
Respondeo: Excellent question. Legally, heart donors have to be declared "brain dead" before they can remove their heart. So the philosophical question is whether a "brain dead" patient is really dead, which means, basically, that their brains are necrotic (i.e., "rotten") and hence there is no possibility of them ever recuperating or even regaining consciousness. (This is what it should mean, although doctors often declare, as 'brain dead', patients who only show some signs of brain dysfunction, and few ever really ensure that the whole brain is necrotic). In any case, brain death is different from a coma (or 'Persistent Vegetative State'), because in a coma, the patient has their brain intact, but it just doesn't work; and comatose patients do sometimes regain consciousness.
The Church has taught that one may hold the opinion that brain dead patients are really dead (i.e., their souls have left their bodies). So if a philosopher or theologian or physician holds that opinion---and most do---they are not heretics for doing so. But the other opinion, namely, that brain dead patients are really alive (i.e., their souls are still informing their bodies), is also perfectly admissible. We are also allowed to hold it. So the issue is open to debate among Catholics.
My own opinion is the latter opinion, namely, that "brain death" is not really death. This is not a very popular opinion, in particular because it means that heart transplants involve murdering a truly living (though "brain dead") patient. A good book that defends this position is Mattei, ed., Finis Vitae: Is Brain Death Still Life?, from which I have quoted before. My reasoning is that there are many biological functions going on in the body of the "brain dead" patient, which means that that patient has to have a soul. The patient is definitely alive. Any biologist (as opposed to a medical doctor) would agree that there is life in that organism. For a formal version of my argument, see this previous post.
Some Thomists would reply that indeed such a patient is alive, but it is no longer a human being, because it does not have a brain. So it is some other kind of living organism, one that is not human, and thus it is permissible for us to kill it.
My reply: the only way that that patient is not a living human being is if, for some reason, the human soul has left the body and some other, (animal or plant) soul has come in its stead. This is highly unlikely (why would God do such a thing?), and, therefore, we should err on the side of caution, and assume it is a human being, and not kill him just to prolong someone else's life. The end (saving someone else) does not justify the means (probably killing someone, no matter how bad a state they're in).