Answer
In most cases, Yes. But the situation is complex.
Officially, citizens of the UK or Ireland don't need a passport to enter Ireland from the UK, but all other nationalities do. But you need some form of ID to prove you are an Irish or British citizen, and the only form of ID generally accepted are a drivers license or a passport. The drivers license must be a UK or Irish license and you must have been born in the UK or Ireland. This is because the drivers license doesn't state your citizenship, only your place of birth. To use it as evidence of citizenship, you have to have been born in either the UK or Ireland. Technically, only those born in the UK before 1983 can use their license to travel, this is because UK law was changed in 1983 and anyone born after this date is not guarenteed to be a citizen.
In short, you may enter Ireland from the UK without a passport if:
•You are a UK citizen, or
•You are an Irish citizen, and
•You are born in the UK before 1983 or in Ireland, and
•You have a full UK or Irish drivers license
Anyone else needs a passport. If you needed a visa to enter the UK, you need a new Irish visa to enter Ireland.
These rules are enforced in the airports and to a lesser degree in the shipping ports, but hardly at all on the land border between Northern Ireland and the Republic of Ireland. When entering the Republic from Northern Ireland, non-UK and Irish citizens (including all EU nationals) are technically required to register their passport at the nearest Garda station (Police station), but this rule is not enforced at all.
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