HMRC’s guidance is in notice 741A: https://www.gov.uk/guidance/vat-place-of-supply-of-services-notice-741a
The first step is to work out whether your supply is B2B or B2C. This depends on whether your customer makes any business supplies. See section 2.4 – if you’re still uncertain, you may need them to confirm.
Then work out whether any specific place of supply rules apply, or whether you default to the general rule. The only specific rule that I think might apply is in section 9, if your supply is deemed to be B2C.
In summary:
• B2B transaction, place of supply is Ireland
• B2C transaction and section 9 applies, place of supply is Ireland
• B2C transaction and section 9 doesn’t apply, place of supply is UK.
If the place of supply is the UK (the last option above), you would be required to charge UK VAT.
If the place of supply is Ireland (either of the first two options) you don’t have to charge UK VAT – but there may be implications for your customer. A VAT registered customer must pay over VAT as if they were the supplier, and reclaim it as the customer. This is called the reverse charge – see section 5 of the HMRC notice for an explanation of how UK customers do this when receiving services in the UK from foreign suppliers. Obviously the reverse charge can’t be done by a customer that isn’t VAT registered, as they won’t have a VAT return to make the entries on. In the UK, HMRC require such customers to include the value of any supplies where a reverse charge adjustment would be required when assessing whether their turnover exceeds the £85k VAT registration threshold (section 5.7 of HMRC’s guidance). The rules in Ireland appear to be different – the link below is to their guidance. Your customer would need to determine from this whether they were required to register for Irish VAT due to this transaction:
https://www.revenue.ie/en/vat/vat-on-services/exceptions-to-the-general-place-of-supply-rules-for-services/received-services/index.aspx