Motorists resident in Ireland with a UK, including NI driving licence are being advised to exchange that licence for an Irish driving licence before the 31 October 2019. In the event of a no-deal Brexit, the UK will no longer be a Member State and so the UK driving licence will not be recognised. People resident in Ireland will no longer be able to drive on a UK driving licence.
However, legislation exists to allow for the recognition of foreign driving licences for exchange purposes in the Road Traffic Acts. If there is a no deal Brexit, the UK becomes a 3rd country and the potential then exists for arrangements to be made under those Acts. Ireland will be pursuing this option. This may take a little time to complete as it involves a formal agreement and legislation here in Ireland, as well as corresponding steps by the UK.