Terambulan's answer is incorrect - different rules apply to intra-EU trade.
As you have stated, the UK business provides its VAT number to the Irish business. They won't charge Irish VAT on the goods, and the UK business gets an invoice with 0% VAT.
This is allowed only between two EU VAT-registered businesses.
For cross-border trade, the supplier does not charge VAT on the goods, so long as they satisfy certain rules (such as obtaining the purchaser's VAT number and retaining proof that the goods were delivered to another EU country). They may, however, have to complete Intrastat returns which provide details of such exports, if they exceed a certain level.
Since the Irish company does not declare VAT on its supply, there is no VAT for it to reclaim from its government. There is a space on UK VAT returns to note down the value of imports from and exports to other EU countries. I suspect the same is true of RoI VAT paperwork.