My daughters father wants to sign over his rights. Can he do that without someone having to adopt her?

Nope

Nope

And even if he could give up his "rights", that wouldn't terminate his obligation (duty) to pay child support. That obligation is, I suspect, what he is trying to evade. It won't work without adoption.

First where do you live, there may be come countries where this is possible, in the US or Canada no, he can not just sign over his rights.

Now he does not have to use them, he just does not call, he does not ask for visits, so if he wants to sign over his rights, ( he can't) but he just never uses them, and it is the same thing.
Now in a vew cases, where a father was allowed to sign over rights or had them taken away, he still pays child support anyway, so if he does not want to see child, the father can just go away and as long as he pays his check he is fine