The EU sets common standards for driver licensing across all Member States. This means that all Member States observe the same standards for obtaining a driving licence, and issue licences in the same standard format. Under EU law, a person is required to obtain their driving licence from the Member State in which they are normally resident. They may drive on their national licence in other Member States. If they move from one Member State to another, they should exchange their licence for the equivalent licence in their new State of residence. UK driving licence holders may therefore drive in Ireland on their licences, and may exchange their licences for the Irish equivalent if taking up residence here.
Were the UK to leave the EU, any future arrangements would be a matter for negotiation between the EU and the UK in the first instance, and I cannot speculate on what might emerge in such a hypothetical scenario.
The Deputy may wish to know, however, that under international road traffic conventions, holders of foreign licences may drive in Ireland for up to a year on a visitor basis.