If a father signs away his parental rights, does he still have to pay child support?

Relinquishing parental rights is not something  the courts take likely.   Although not always stated the general standard used bu courts is for "good cause."  Relinquishing rights for adoptive purposes is seen in a better light.

The actual process is called "termination."  This can be done voluntarily. You would need to seek legal help in your area with the entire process. 

Typically, terminating your rights relieves you of any obligation to your child from the date of termination forward.  It is doubtful that any jurisdiction would relieve you of past support especially if it can be or is reduced to an order or judgement.