If a parent gives up the rights to a child does the past due balance remain or does the other parent have to sign off on it

Well first, you start with a questionable premise. There is a myth that a parent can just give up their rights to a child. But its only a myth.

Only a court can issue a TPR and courts are very reluctant to do so. They will never issue one just to allow a parent out of paying support. Generally a TPR will only be issued to clear the way for an adoption. In that case, any future support obligation will cease when the adoption becomes final. But the arrears will still be owed. In such a case, though the other parent can agree to waive the balance.

However, if the custodial parent received public assistance. The state may insist on reimbursement.