When you turn up at the Aer Lingus counter in the US, you are presenting a ticket that has you arriving in Ireland as your final destination. As such, you must be able to show the airline that you are entitled to enter Ireland. If you do not have the necessary visa or a passport that allows you to enter Ireland, then Aer Lingus will almost certainly refuse to carry you. Having a separate ticket on the same or next day is beside the point; as far as Aer Lingus are concerned, you are a customer that is flying to Ireland and therefore if you are not able to enter Ireland they cannot legally transport you.

(Not that it really matters, but all arrivals in Dublin - whether transferring to another flight or ending their journey at DUB - have their passports checked)