Does a man have the right to give up his rights to a child/

Only if the mother has a new partner who wants to adopt.

The law normally protects the rights of the child,

In cases without someone to adopt ( step parent) even if rights were given up ( but seldom is allowed) they will still have to pay child support.

So this is easy, just pay your support, ( that is a obligation, not a right) and don't use any right

Like Chuck said - you don't have to use your rights. You just have to pay child support.

For what reason would you want to give up your rights?