My fiancé wants to adopt my daughter. I want to know that if I do decide to bring my daughter's father to court to give up his rights and he doesn't show for the court date.. will the court automatically take his rights away?

Courts will not do anything automatically, if he does not show up, your attorney makes a motion to continue with the cause and rule in your favor.

The judge most likely will, but can do several things. With judges and courts there is seldom a automatically.