Dingle is open year-round but functions very differently between seasons. May to September is the proper season: the days are long (it doesn't get fully dark until eleven in June), the boat trips to the Blasket Islands run reliably, the music is in every pub, and the town is busy without being heaving.
July and August are the peak — accommodation prices double, the Slea Head Drive can queue up behind tour coaches, and you need to book restaurants a few days ahead. The Dingle Races in early August and the Dingle Regatta in mid-August are worth timing a visit around, with most of the town turning out.
October to April is much quieter. Many of the smaller hotels, restaurants, and shops close, the ferry to Great Blasket Island stops, and the weather is famously variable. But the town stays alive on the back of the locals and the year-round pubs, and the peninsula scenery is in some ways better in moody winter weather. St Patrick's weekend in mid-March is the big winter festival.
Bring waterproofs whatever the forecast. Atlantic weather changes on a fifteen-minute cycle.