the rules for business to business (B2B) supplies of services in the EU changed from 1 January this year. From this date, almost all B2B supplies of services are zero rated by the supplier and instead self-accounted for by the customer, provided the customer is registered for VAT in their own country.
In this case, it would mean the Italian company zero rate their invoice to the Irish company but quote the Irish company's VAT number on their invoice. Then the Irish company effectively charge themselves Irish VAT on the invoice (and reclaim it if they are entitled to do so).
There are a number of exceptions to this rule but they generally related to work on land and buildings. If the Italian company are, correctly, entitled to charge the VAT (and I don't think they are) then the Italian VAT can be reclaimed through the 8th directive procedure. The administration of this changed this year too and the reclaim can be processed through the Irish Revenue's website.