The position is that a non-national must first obtain Irish citizenship before being eligible to hold an Irish passport. I have set out below some general information about the various ways a non-national can acquire Irish citizenship.
A person whose father or mother was an Irish citizen at the time of his/her birth is an Irish citizen from birth. A person whose grandparent was born in Ireland may become an Irish citizen by registering in the foreign births register at an Irish embassy or consular office or at the Department of Foreign Affairs. A person whose great-grandparent was born in Ireland may register for Irish citizenship provided that their parent had registered in the foreign births register by the time of the person's birth.