I'm not sure you can make a case against it if you don't include God, so I don't think I can convince you, Jo.(And indeed, I know practically nothing about the subject, so I will only be able to provide a basic argument, and won't be able to defend it very well.)
The basic argument is "What right do you have to sell your body?"(or parts of it) You did nothing to earn it; it was entrusted to you by God, and He will demand an account of it from you someday, as He will all your gifts and talents. You don't have the right to take your own life, because your life is not your own, but God's; so for the same reason, you don't have the right to sell your own body, for it belongs to God. Of course, that leaves the question of the sale of hair, but I suppose we could make a distinction because donated organs are still living, and hair, especially after it is cut off, is defiantly dead.
Then there is the question of price. Some things are priceless, in a very true way. Nothing in the world could buy them, no amount of money or goods could equal them in worth, because of the inherent dignity they posses. They can only be given out as a free gift. To put a price on them is to insult them highly.
The sale of sacred things has always been rightly regarded as sacrilegious. It is an insult to them. For example, the sale of indulgences(
) is an insult to the very nature of an indulgence(
), a sacred gift of God. To buy and sell it is to insult God's gift. Just so, the human body is a sacred gift of God. The body is, as Lewis said, coursing with the energy that made the universe; and to try and put a merely human price on it is a degradation of it's great dignity and worth.
Those are some of the reasons religious people will cite to condemn the sale of organs. There are other, practical reasons as well. I found , which lists a few of them.