What responsibilities do you lose when relinqishing parental rights?

You don't lose any responsibility. You merely lose the right to see the child, have anything to do with the child's up bringing. So you lose the resonsiblity of taking them to ball games, changing diapers, and tucking them in late at night

You still have all the same obligations, you will have to pay child support, and also insurance if it is court ordered.

That depends, are you relinquishing for purposes of placing a child for adoption?

HOUSE BILL NO. 160 - Parental rights, terminated, when (http://www3.state.id.us/oasis/2003/H0160.html).
Maybe this will help.
As for myself I am still confused. That is why I can not make a definite answer to your and others questions about this.