If the vehicle is brand new then you tell the dealer that the car is for export to Ireland, you will sign a declaration stating that the car is for export and the VAT will be paid in Ireland, this will also be signed by the dealer, who will issue the vehicle with a number which has X (export) on the plate so that the UK customs can tell. This declaration is then sent to the DVLA in England. The dealer will give you the V5 registration document to present at the VRO Office.
When you enter Ireland you will have to pay the Irish Vat on the purchase price of the vehicle converted to euro (net of vat price). Ìn addition, VRT will also need to be paid.
You may not claim the Vat back on a passenger vehicle.
"NOT ADVISED": I live in the North East and what many people do is buy a new car in England and as above tell the dealer its for export, export the car to ireland do not register it till its either 6 months old or has done 6000 miles, after either of these dates, which ever comes first, people are then registering the car and not paying any vat.
"NOT ADVISED": I also am aware of cases in which people have claimed the Vat back on passenger vehicles. Although incorrect they believe that they will not have a vat audit and get away with it. If they do have a Vat audit and it was discovered that they incorrectly calmed back the Vat on a passenger vehicle, the Vat will have to be paid to the Revenue in addition with Interest and Penalty charges.