Game of Thrones is filmed in various places worldwide, not just Ireland. This map gives you a good breakdown.
Many of the castles and other locations are actually filmed in several different locations combined with computer graphics, so don't expect to be able to find something exactly like what you see in the show.
In Ireland there are two castles that stand-in for Winterfell and you can tour them with Game of Thrones tours as mentioned in the answer to this question: Game of Thrones in Dublin. For Castle Ward you cannot stay in the castle, but you can stay nearby and there's are GoT themed activities and tours. The other place used is Carncastle, but it's more a place than a castle and appears to have been used for scenery and other establishing shots.
Doune Castle is Scotland was used as Winterfell in the pilot, you can tour that but not stay there.
The twins appear to be filmed in Moneyglass Estate, County Antrim but I think they're largely computer generated since I can find nothing that looks like them.
The other place on the map in Ireland that is used for castle filming is Magheramorne Quarry, standing in for Castle Black and Hardholme. However this is a constructed set and not a real castle (although that page says they've built a fair amount of stuff). I can't find tours there (since it's an active filming site I'm not sure there would be) and you certainly can't sleep there.
The Eyrie is inspired by monasteries in Metora in Greece. You can visit them but it doesn't appear you can stay in them.
As far as I can see there's no way to stay in anything remotely like a castle from the series. The closest would be places such as Morocco or Dubrovnik in Croatia which is where a lot of Kings Landing has been filmed. I'd imagine you could find somewhere in one of those places that would be the closest you can actually get.
There are many castles which might provoke the same sort of feeling and experience which you can stay in (see this collection on Pinterest for a start). But no way to stay in an actual castle from the series -- primarily because they're fictional places cleverly created from real world places, purpose built sets and computer graphics.