Yes, a UK VAT registered recipient should use the Reverse Charge. But the Irish supplier should not charge you VAT - did you provide your UK VAT number to them?
As I understand it, the point of reverse charge is that the EU company does not charge VAT to you. You have to account for the VAT instead. You pay it in the UK, instead of the Irish supplier paying it to his government. For a VAT registered business, this means you would account for it as sales VAT, but then also, as purchase VAT. Its net value will appear in Box 6 and Box 7. (I trust its EU, and not Northern Ireland?)
It sounds like reverse charge is not relevant, as the Irish invoice would show no VAT, and detail a disclosure about you needing to account for reverse charge, (directive 244, I think) - either that or the Irish supplier is doing things wrong. Place of supply usually determines how the VAT is accounted for.
If you pass your VAT number to the Irish supplier, this should stop any further VAT being charged.
