If the custodial parent in a court of law relinquished custody of a child to another person; does the non-custodial parent continue to pay child support to the parent that has relinquished custody?

If the parent that has relinquished rights of a child and someone else is caring for him/her then I don't believe that parent would continue to receive child support. Why would they, they don't have the child any longer ?

Well first you have several issues, you have two parents, one has custody, the other should have visitation rights and obligation to pay child support. So if the one parent is going to give up custody, that custody should go to the other parent, not to some third party, unlesss the other parent agrees. I could not see a court agreeing to this unless all parties have agreed.

The child support would not go to the parent who gave up custody, but you will have to go and have the child support ordered modified to pay the new person with custody, this should have been done with the custody of the child was changed in court

Unless your court order has changed you still have to pay. If the custodial parent relinqueshed custody and the custody papers changed then the support order would have also changed. So (assuming you are the non-custodial parent) if it was just an "informal" change of custody you need to go to court and file a petition for custody on the grounds that the custodial parent violated the order or ask to have the payments remited to the new custodial parent.

BTW for the record if both were to agree on giving up the child both parties are responsible for child support. The obligation doesn't stop just the person who ultimatly gets the funds is all. If its taken from your check and given to a state agency then that's not going to change.