Why should organ donors be tested?

Because the donor might be suffering from a medical condition which would render the organ useless to the recipient, or which would infect the recipient with a disease which would cause illness or rejection of the organ.

Transplant patients are at high risk of infection and other complications even with the best possible precautions and organ transplant is a delicate, lengthy and costly procedure; there is no point performing such a procedure if the donor organ might carry with it a health risk which could negate the entire process.