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.
travel.stackexchange.com/users/49829/Conrad
The fact is that all British and Irish citizens have an unconditional right to enter each other's countries.
Meaning anyone with documents proving or making it plausible that they're British/Irish must be let in.
As such, if the child is yours, their birth certificate stating your relationship is enough, although some check-in handling agents (if flying) may not accept it (contact them beforehand and ask)
If it's not your child, ask the parents to obtain the child's birth certificate as well as a legalised/notarised affidavit of consent containing a copy of their ID as well as their contact details.
travel.stackexchange.com/users/39782/Crazydre
Be careful though.
Ryanair won't accept anything other than a passport between Ireland and the UK.
As far as I am aware, they are the only carrier with this requirement.
I am not clear how this affects accompanied minors flying on this route with Ryanair, though.
Note also, based on experience travelling using only a driver's license, that you may be asked to produce your boarding card as evidence of your journey, this has happened to me several times recently, when entering Ireland, so be sure to keep this handy.
travel.stackexchange.com/users/73215/ajay
RYANAIR ONLY ACCEPT PASSPORTS.
I've just done a live chat to see if I can take my 10 month old baby from the UK to dublin with a birth certificate and they have said they will only accept passports.
travel.stackexchange.com/users/94856/Lucy Rowley
From my own experience as both a bus passenger and travelling via private car as a passenger, the experience differs .
My ID was always checked before boarding at Pembroke Dock as a bus passenger and randomly as a car passenger at Rosslare.
It seems to depend on the mood of the staff and how busy they are.
I have not got a passport yet , but no doubt will need to get one sooner or later.
Have never been asked for ID for our child of whom we have Special Guardianship.
travel.stackexchange.com/users/104002/Sandra NEATE
Where in the UK are you?
If you travel by boat, you won't have your ID checked, so another option is to drive, or Rail and Sail.
travel.stackexchange.com/users/16657/JMK
I understand from various answers here, and from .gov.uk that I don't need a passport to travel to Ireland: I can just show my UK driver's license.  But what if I'm travelling with a 10 year old child who does not have a passport, and obviously no driving license either.  Is there some other form of ID which would be accepted?
UK citizens traveling to Ireland with a child having no passport, what's the alternative for the ID?
travel.stackexchange.com
<customs-and-immigration><paperwork><uk-citizens><ireland>
travel.stackexchange.com/questions/109061
travel.stackexchange.com/users/56792/Stewart
British and Irish citizens under the age of 18 do not need a passport when travelling with a parent or guardian, but proof of legal guardianship such as a birth certificate should be provided. If you are not the parent, you should also bring a legalised/notarised affidavit of consent from the parents containing a copy of their ID as well as their contact details. Children of any other nationality are required to hold a passport that is valid for the duration of their stay. In most cases the minor does not require a form of ID unless they are a none british citizen or flying with Ryanair. It is the parent/guardian who requires a valid form of ID and proof that the child is there's or that they are the legal guardian. This may come in the form of a birth certificate and/or a legalised/notarised affidavit of consent which would also contain a copy of the real parents ID if the child is not yours.
Provided that the child is a british citizen then they do not require a passport or driving license in most cases. It is the parent/guardian who re-quires a valid form of ID as well as some form of proof that the child is either theirs or they are the legal guardian. If the child is not a british citizen then they require a passport.  S1: Ryanair won't accept anything other than a passport between Ireland and the UK even if the passenger is a minor. If traveling by land or sea you do not have to provide any form of ID.