Dual citizenship is not an issue any more under US law, but US immigration officers are not lawyers. They may not know the intricacies. Be careful with them.
Before 1990, the US made it very hard to retain US citizenship if you VOLUNTARILY took another citizenship. That would not apply in the case of your partner, Cludet, because he would be deemed Irish from birth, but the last thing you want is to raise questions with Homeland Security.
Also - you can still get into hot water by not using your US passport when entering the US.
Obviously if they specifically ask you for some reason whether you have any other citizenship, I wouldn't lie about it. When I applied for my new US passport at the consulate a couple of years ago, they asked me whether I had taken any foreign citizenship in the meantime, and I told them yes, UK citizenship, and they said that's ok, we don't need to see the passport. So generally speaking it's a non-issue, but I have heard some horror stories lately about US immigration officers going postal, even on Europeans.