The UK's first altruistic liver transplant (that is, one from a live donor to a stranger) was performed last month. The unusual thing about the liver is that one can donate a liver lobe and then one's liver can re-grow so, unlike donating a kidney, one may be no worse off in the long term. In the short term, however, the donor may suffer as a result of the operation, which in this case took four hours and left a 6" scar, as well as the risk of infection and/or psychological problems. This pieces also highlights the risk of death: approximately 0.5%.
Given these costs, it's unsurprising that Dr Aluvihare - transplant specialist at King's College Hospital - is quoted as saying "I personally have some reservations about altruistic donations. I believe if we did everything we can to improve the supply of donations after death we wouldn't have a need for this type of donation". So the case for favouring posthumous donation isn't simply that people's organs are of little use to them after they die, but also that the costs of removing those organs are much lower.