You will not need a visa if your going on holiday or a short break. UK passport holders will not need a visa to travel to the EU for stays of up to 90 days in any 180-day period. The same applies vice versa: holders of Irish and EU member state passports will not need a visa to travel to the UK.
You may need a visa or permit to stay for longer, to work or study, or for business travel. Further information can be found here.