If the biological father has no rights do you need his permission when adopting the child when marrying the biological mother?
You need to see a lawyer. Look for "Legal Aid" in your state as
they offer free or low-cost legal advice.
If someone is the "biological" father, he doesn't adopt the
child, it is his child. He is legally obligated (in the USA) to
provide support (money) for the child's needs. The only exception
is if the mother was married to another man at the time of the
birth, then legally the husband is considered the baby's father
(even if everyone knows and admits that the other man is the baby's
biological father).
If you're asking whether the biological father can be forced to
take care of the child in his own house, the answer is "no" he can
give up his parental rights to the child.
If you're asking whether the biological father can take the
child away from the mother and her boyfriend/husband, the answer is
"maybe" if he can PROVE to a court that the baby's mother is unfit
and he (the biological father) is a better parent.
If the father WANTS to be involved in the baby's life even
though the mother has a new boyfriend/husband, the father MUST be
allowed to have visitation with his child. Not allowing him to see
the child may cause the courts to consider the mother not fit to
raise the child.
Again, GET LEGAL ADVICE from a lawyer.