I wouldn't take Thurman as being any definitive authority for Tibetan Buddhism but I think he may reflect a current view on this thing of organ donation......http://www.donatelifeny.org/pdf/RobertThurman.pdf
So far at least, none here is advocating for being a receipant of donated organs but for offering them to people in need.
As has been pointed out..... if anything such offering for free, if widespread enough, would largly eliminate the illegal trade in organs which certainly exists.
Hearts i know not of...one dharma master of another tradition has just seemingly received a donated liver.
The brain may be clinically dead and the heart still beating that is medical fact upon donation.
As brain is dead with no possibility of furthur brain activity there seems no possibility of advanceing any furthur in the spiritual as result of having this human body at that time, other than as a strict function of utilization of karmic effect. A beating heart with no brain function would equilivent to many animals. So one would for all intents and purposes be a animal, in a animal realm. That is fine and may remove some latent karmic effect but someone else may need the heart and has asked for it.
So thusly as the individual will not advance as human as result of having no brain but a heart there is no fault in giving the heart for another to live. They, that person, may or may not advance in the dharma but it then becomes possible that they may. Certainly if one is a base corse person presently.... having death at that time will virtually assure them a bad rebirth. If they get a new heart they may have a slight chance of changing their ways, and obtaining a more fortunate rebirth.
So it is better to give the heart if brain is dead if asked for. If not asked for, it, the situation of being brain dead but still alive with beating heart, may remove stored karma that has necessitated this occur, and it should not be done.
I have heard of HHDL speaking very highly of one monk who offered to give not only one eye but two eyes to someone who had become blind. So while he has not spoken to the issue exactly perhaps, I suggest thurmans comments follow through on this thing. Though that monk would remain alive it is still a organ donation.
"This order considers that progress can be achieved more rapidly during a single month of self-transformation through terrifying conditions in rough terrain and in "the abode of harmful forces" than through meditating for a period of three years in towns and monasteries"....Takpo Tashi Namgyal.