One of our Ireland options for next Spring is June 23 thru July 1, visiting the Dingle peninsula, the Ring of Kerry, and the Beara peninsula. How crowded with tourists is it likely to be then?
Let's just say you'll have plenty of company at most of the major tourist attractions.
If you have a car, you can mitigate things somewhat by going early (I'd say before 10 AM) or later in the afternoon to avoid the tour buses.
Slea Head Drive should be OK if you get a reasonably early start. Dingle Town can fill up quickly at midday when the buses arrive, but gets mellow again in a hurry after they depart in the afternoon.
Beara shouldn't be a problem - most itineraries don't include it and the tour companies don't go there (as of our visit last summer anyway). Might head out early if you want to catch the cable car over to Dursey Island - it's capacity is so small that lines can form quickly.
If you're self driving the ROK the same advice applies - go early or late. A nice side trip is the spur that goes out to Valencia Island and the scenic little town of Portmagee...where the Skellig boats depart from. The buses don't go there either so it makes for a much more relaxing experience.
If you don't have a car, well....