Its kinda hard for him to be allowed to sign over his rights. The judge wont allow that usually unless there is a stepparent willing to adopt the child. If he does succeed in signing over his rights, he will not be responsible for child support. He is the childs father and he can fight you for custody. He may be awarded custody of the child and you will have to pay him. I would recommend you do nothing for right now. Wait until the child is 2 or so, and make sure you have evidence that boyfriend has not been an active father and has not had much involvement in the childs life.
Do not put him on the birth certificate. Hes probably not going to take you to court so I would do nothing for right now and hope he goes away. Maybe move far away from him. (Youre not married, so you are allowed to take the child and go.) Then when he hasn't been around for the kid and knows nothing about it, sue for child support. You can be awarded for 2 years retroactively and you will have proof he hasn't been around. He will get visitation (every other weekend or something like that) but if he hasn't been involved and doesn't know the child he probably won't use it.