Lets say you have two passports. One Irish and one Russian. If you use Russian passport to enter Iraq, will you still have trouble with ESTA with your Irish passport?
You're asked to fill in a questionnaire when entering the US as a foreigner. I think lying at the 'did you visit...' questions would be grounds for the US to declare you 'persona non grata', so it's not a risk you'd want to take if want to keep the possibility of entering the US again.