I have had varied experiences. First neither are part of the Schengen Agreement, so each country technically handles it's own Border Controls. The UK and Ireland do however have an agreement commonly called the Common Travel Area (CTA) in which many border checks between them were eliminated. That said, on a trip in the early 2000's (2004 I believe) I did have an immigration check when arriving by air at the Dublin airport from Leeds Airport, and given a month or so stay in my passport (US Citizen, told them I would be there for a week). I do not believe that I had any immigration on return to Leeds though, can't recall. I was flying on RyanAir, heard later that the check was something prompted by them, not sure if that is true, but the guy stamping my passport was Border Control, not an airline employee. It may have been the terminal we arrived in as well, they may have used the international gates as opposed to domestic for convenience. There were changes to the agreement since then (2008 and 2011) so my experience may just be due to the time period.
I believe that the CTA also primarily pertains to UK and Irish citizens, not necessarily foreign travelers, so the UK people on here likely will say no border controls, North Americans may have other experiences.