Ireland does not have a dual citizenship agreement with any country. There is nothing in Irish legislation to prevent an Irish citizen holding simultaneously citizenship of another country or countries.
The laws governing citizenship and eligibility for passports of other countries are a matter for determination by those countries. Where a citizen of another country is eligible and applies for Irish citizenship, he or she would need to inquire from the authorities of that country about the effect, if any, such an application would have on their original citizenship. The Irish passport authorities have no difficulty with Irish citizens holding Irish passports as well as passports of their other country or countries of citizenship.