The Irish authorities don't require UK citizens to provide a passport but all airlines insist upon a suitable form of identification. All airlines will accept an EU (photocard) driving licence as a valid for of ID for travel within the UK or between the UK and Ireland. Ryanair
only accepts a passport or driving licence; other airlines are more flexible. (The only exception with Ryanair is for under-16s, travelling with an adult, who do not require ID).
Ryanair's policy:
http://www.ryanair.com/site/EN/faqs.php?sect=p id&quest=photoid
easyJet will accept
any form of photo ID (such as a student card or bus pass) as long as it bears your name and it's still in date. (I can't provide a web link to show that. I've just phoned their 'customer experience centre' to check)
Chris