(In this situation, getting an Irish passport = the British version of marrying a Canadian if Donald Trump wins.)
Northern Ireland also voted strongly for Remain, largely because of the supporting role the EU played in its peace process, and because it's the only part of the UK to have land borders with the EU (with the Republic of Ireland).
It turns out that century-long violent disputes that have only recently reached a wary peace don't respond well to potentially having militarised borders added into the mix, especially when people cross them every day to go to work. More here – it is complicated.