You may be lucky and not need it, depending on your route and travel mode. But the safest is to travel with a passport, even if you are an Irish or UK citizen. A better-than-nothing option is to at least have a national ID card if you are an EU citizen as it's accepted in EU countries, including The Republic of Ireland, but the UK is no longer required or expected to accept it.
Note that the Common Travel Area is intended for Irish and UK citizens:
Common Travel Area rights can only be exercised by citizens of Ireland
and the UK. If you are not a citizen of Ireland or the UK, you cannot
exercise Common Travel Area rights.
For example, having a valid visa for the UK, technically does not mean that one can travel freely to Ireland. They also say:
There are no routine passport controls in operation for Irish and UK
citizens travelling between the 2 countries.
However, you must show identification to board a ferry or an airplane,
and some airlines and sea carriers only accept a passport as valid
identification. You may also be asked by an immigration officer to
prove that you are a citizen of Ireland or the UK, so you should carry
a passport with you. You can also use an Irish passport card, or other
proof that you are an Irish citizen.
In practice this means that if your form of identification is accepted for you to get on a plane from the UK to Ireland, and the plane takes you to a disembarking point where only travelers from the UK are expected, there may not be a border control at all. The reality of some airports is that sometimes travelers may be all lumped together from different flights, which means they all have go through passport control to enter the country.