You need to complete a Dividend Withholding Tax (DWT) form from the Irish Revenue - http://www.revenue.ie/en/tax/dwt/forms/index.html. Look for the "Non-Resident Form V2A - Exemption from DWT for A Qualifying Non-Resident Individual in respect of relevant distributions".
UK residents - claiming tax relief or exemption for foreign income or gains HMRC: dta-claim.html.
DT9882 - Double Taxation Relief Manual: Guidance by country: Ireland: Dividends DT9882.htm.
Note that you cannot ordinarily claim exemption from paying Irish tax on your share dividends. This is still the case even if you hold your shares in an ISA.
None of the information I give here constitutes advice, and may not be correct for your own circumstances. You should always consult a tax professional, and don't just believe anything you read on the Internet.