Here the childs welfare is paramount, so men who are not even the father, and the child is a result of adultery, can be forced to pay child support, while the actual father does not. Kooky stuff.
Actually that's how it works in the UK too, estranged lesbian couples notwithstanding. If you're the biological father of a child or you've officially adopted them you have to pay child support, even if the child was conceived out of wedlock. That includes adultery, one night stands etc.
It's not unheard of for a man to raise a child as his own from birth without having any legal obligation to pay child support if he and the mother were to separate. If one were to take a rather callous view one could argue that a man shouldn't have an obligation to support a child he wasn't responsible for bringing into the world. A family court might order differently but that's the way the official child support scheme works in the UK and I fancy that such an order would be on shaky legal ground. (In the UK child support is handled by either the courts or a separate government agency; from what I've heard only the family courts deal with it in the US.)
In the assisted fertilisation example you gave the donor would not have to pay as they aren't legally the child's parent, unless the "donation" was made the old-fashioned way.