For VAT purposes it is the physical movement of goods that is relevant, rather than the invoicing name or address. An invoice to an Irish customer when the goods are not leaving the UK will not normally be classed as an export, and UK VAT would be chargeable. However, goods being sent to Ireland on behalf of a UK customer without an Irish VAT number are still subject to UK VAT at the usual rate (even if the customer is registered for VAT in the UK).
A VAT-registered UK business does not need to charge VAT on goods it is sending to Ireland, provided it keeps documentary proof of export. It must also obtain the Irish customer’s VAT registration number and show it on the invoice. If the goods will finally get exported to Ireland then you need proof of export.
In addition, the net value of the sale must still be reported on the UK VAT return, in boxes 6 and 8. It should also be included on the EC Sales List, and the Intrastat return if the business is required to make one.
The UK business is able to reclaim any UK VAT on the goods which it is exporting, subject to the normal rules about reclaiming input VAT.
If the Irish customer is not registered for VAT, and the UK business is not registered for Irish VAT, then UK VAT must be charged. In this case, the sale is included in box 1 and box 6 of the UK VAT return, but not box 8. It does not get included on the EC Sales List, but is liable to be reported on Intrastat. No VAT is chargeable on goods that would normally be zero-rated or exempt when supplied in the UK (for example, books, children’s clothing and some food items).
Hope this helps