For organs such as liver, kidneys, heart and lungs the 'donor' must first be declared 'brain dead' (which is not death, it is a made up term used for the sole purpose of procuring organs). Then, they will be taken to surgery (usually without an anaesthetic.......they just get paralysing agents to stop them flinching as they are being cut up) and one by one their organs will be cut out of them until finally their 'beating' heart is cut out. At that point, the donor dies.
Nobody would take an organ from a truly dead person and consider it to be of any use to anybody else. Organs are only useful if they come from a living human being whose life has been judged as less valuable than the one their organs are going to go to. Not forgetting of course, that the 'brain dead' person will make a lot of money for hospitals instead of having to have a lot of money spent on them. Always follow the money.