You can hire a car on mainland UK and take it to Ireland. Alamo and Europcar will allow you to do it. If you are taking the car to Northern Ireland, then there is no charge for doing so. If you are taking the car to the Republic of Ireland they will charge you and additional fee. Last time I check the cost was about GBP 50.00 extra. This covers the cost of getting someone like the AA breakdown service retrieving the car from Ireland in a case where you are not able to drive it back.
Taking a hire car to Ireland