This question comes up every few weeks!
The Irish authorities operate an 'open border' policy on all flight from the UK, so you definitely don't need a passport on arrival. Where you might need a passport, however, is on departure. This is because, under anti-terrorism legislation, airlines are required to confirm the identity of passengers before permitting them to travel. Each airline will have its own rules as to what type of identification is required when you check in, so you need to consult the airline (or see their website) to check their current rules. Ryanair, for example require you to possess one of the following:
(a) a passport ; or
(b) a national identity card (obviously this only applies to citizens of countries that have ID cards) ; or
(c) a European Driving Licence (This means the 'modern' type of licence used within the UK, which includes a photograph).
http://www.ryanair.com/site/EN/faqs.php?sect=pid&que st=photoid
Chris