Many countries that allow dual citizenship and hence dual passports, require that in order for the dual citizenship to be approved and remain in force, the holder of the passport must enter his country of original citizenship (I'm assuming OP was a Canadian citizen before getting Irish citizenship), with the passport of the country of original citizenship and cannot enter with the passport of the "new"/"second" citizenship.
The example I have encountered was a South African obtaining Canadian citizenship later in his life and then trying to visit South Africa on his Canadian passport. He was refused entry as he held South African citizenship and needed to present his South African passport.
Cannot vouch for Canada or other countries, but I'd advise the OP to check with the local embassy.