If your annual income is under about $80,000 you don't have to pay taxes to the US. You're still required to file, but you just fill out a form 2555 indicating your Irish income (no evidence necessary) and you're fine. This has nothing to do with bilateral agreements - it's standard for US citizens living abroad.
I don't know about any such agreements that might apply in the event you go over that threshold, or about capital gains.
I always file my US tax returns, but I've known a few dual citizens resident abroad who don't bother, and they've never had a problem. Unless you're stinkingly rich the IRS probably isn't going to bother with you.