No. UK & Rep of Ireland have had a "no passport" agreement for decades. However, if you are flying, some airlines may insist on a passport for ID purposes. If going by ferry, you will have no need to carry one.
Assuming of course that you are a UK citizen?
>>>Most airlines insist on a passport if you fly within the UK
Wrong! Ryanair is currently the only airline that requires a passport for travel within the UK (or for travel between the UK and Ireland). All other airlines also accept alternative forms of photographic ID, such as a driving licence or even (in most cases) a bus pass.
Ferry companies don't require passengers to carry passports for travel between the UK and Ireland but they advise carrying some form of photographic ID.
The above applies to British and Irish citizens. Nationals of other countries theoretically do require passports for such travel but there seem to be very few checks anyway.