Father Signing Away his rights, but the mother doesn't agree

He can petition the court to terminate his parental relationship. This is covered in § 161.103 of the Texas State Statutes.

It would be up to you to contest the petition (possibly with the claim that terminating would not be in the best interest of the children but, if he claims it would be you might be obligated to prove that it wouldn't be.) and then the judge would decide.

If he thinks he can terminate the relationship to get out of paying child support he'll probbaly find out otherwise. His petition could be granted and he could still be ordered to pay child support.

If he files you need to find a lawyer in your area that deals with family law. You can call the Bar Assoc. (look in the yellow pages!) in your area and get a referral to one.

I would think so! He can relinquish his parental rights
but would need to continue paying child support.

Frankly Ikean, if he wants to sever all emotional ties
to his children, then let him do so. It is far more harmful to
the children if they're disappointed over and over again
by him for not showing up and not being interested,
as if he leaves alltogether and won't be part of their
lives at all.

How much contact does he have with the kids and how much contact do they want? I guess the age of the children matters as it is easier to explain things to a 14 year old than a 3 year old.

Also, why does he want to sign his rights away? Usually it's because guys want to get out of financial support or because there's a new woman in the picture who doesn't like the kids.

I guess it comes down to: do you want a person in your children's life who doesn't want to be there?