I found this article (by
Ven. Bhikkhu Bodhi) recently when I was looking for a description of "compassion".
It says (beware that the following quote is taken slightly out of context),
Lay people support the Sangha with their simple material needs of food, robes, dwellings, and medicines. They also give generously to the poor and disadvantaged. In Sri Lanka, for example, blood donation campaigns are common on Buddhist holidays, and many people donate their eyes to eye banks and their bodily organs for medical research after their death. I learned recently that in Sri Lanka, more than 200 monks have donated kidneys, without any thought of remuneration or any other personal benefit, solely for the privilege of giving a bodily organ.
I infer that the "kidney donation" is from living donors.
In summary, I think that Theravada Buddhism says it's okay to do it (perhaps with some limitations which I hope are obvious regarding harmlessness).
This Buddhism and organ donation page from the UK claims that it's ethically okay, for example here is one of several quotes,
Organ donation is acceptable in Theravada Buddhism. It is a Buddhist virtue to generously extend help to other sentient beings and this covers the case of organ donation." (Phramaha Laow Panyasiri, Abbot, The Buddhavihara Temple)
It also says,
There are no injunctions in Buddhism for or against organ donation. The death process of an individual is viewed as a very important time that should be treated with the greatest care and respect. In some traditions, the moment of death is defined according to criteria which differ from those of modern Western medicine, and there are differing views as to the acceptability of organ transplantation.
I tried to ask about "the moment of death" in this separate question: How long to leave a body undisturbed after death?