I'm a non-EEA national, a spouse of EEA, and I hold a residence permit under the article 10 2004/38/ce, we entered Ireland without any problems, my question is, do I have to apply for an Irish Residence Permit?
Yes, you must apply for an Irish residence card (Article 10) to replace your Italian residence card if your spouse intends to stay longer than 3 months.
EU Treaty Rights - Immigration Service Delivery (Ireland)
Freedom of movement
You cannot use a ‘residence card of a family member of a union citizen’ that has been issued by another member state to authorise your long-term stay in Ireland. Your residence card must be issued by the Irish authorities.