The child would not need a passport, but you must be able to prove that you are their legal guardian.
Assuming that you are flying, the best thing to do would be consult your airline's webpage, as sometimes policies can vary between airlines.
For example, from Flybe:
Republic of Ireland travel
British and Irish citizens must have a form of photographic ID which proves their nationality/citizenship for immigration purposes e.g. a valid passport or full/provisional photographic driving licence.
British and Irish citizens under the age of 18 do not need a passport when travelling with a parent or guardian but it is highly recommended that they have one. Without a passport, the parent/guardian must be able to prove the relationship with the child, showing birth certificate, marriage certificate etc.
A child of any other nationality must hold a passport which is covered for the period of intended stay.