If an unmarried father signs the birth certificate, is he allowed to pick up the child whenever he wants despite the mothers wishes?

Is there an arrangement that you already have with him? You really need to go to court and get the visitation nailed down or the insanity will continue. By signing the Birth Certificate he is the legal father of the child.

Absent a custody and visitaiton order, in theory, yes.

Absent of any court order both parents have equal rights. However, if they do not live together, the parent the child lives with is in control. So yes, the mother can deny visitation.

So the father should go to court to establish custody and visitation, particularly a visitation schedule.

... the mother can deny visitation.. .
"Can", as opposed to "may".

Let's clarify here: either parent would have the right to take the child. But that's why I said "in theory". The parent seeking to have the child (the father, for example) could go to the other parent and say "I want to take our child with me." But if she say's "No", what's he going to do? Answer is that he would have to sue in court..

No. There is no agreement as of now. The child lives with me. (the mother)

If you deny the father visitation rights, when he takes you to court, the court will not look kindly on it.

You are better off filing for custody before he does.