June is a huge month for festivals. Most notably, this is the month when the Venice Biennale begins (every other year, on odd-numbered years). Also note that June 2, Republic Day, is a national holiday, so many businesses, including museums and restaurants, will be closed..
June 2 - Republic Day (Festa della Repubblica). This big national holiday is akin to Independence Days in other countries. It commemorates Italy becoming a Republic in 1946 following the end of World War II. Look for posters announcing special concerts and festivities around town.
Early June (every other year in odd-numbered years) - La Biennale. The months-long contemporary arts extravaganza that is the Venice Biennale begins in June every other year during odd-numbered years and runs through November. Read more about the Venice Biennale.
Early June (every four years) - Palio of the Four Ancient Maritime Republics. If you want to witness a boat race rife with medieval pageantry, look for the Palio of the Four Ancient Maritime Republics, which Venice hosts every four years. Il Palio delle Quattro Antiche Repubbliche Marinare is an annual traditional regatta that shifts locations amongst the four ancient maritime republics: Venice, Genoa, Amalfi, and Pisa. Preceding the boating competition is a parade, in which participants don medieval dress to march through the streets, complete with flag bearers, horses, drummers, and trumpeteers. Venice is set to host the Palio in 2011, 2015, 2019, and so on. See 2009 procession photos taken in Pisa.
Early- to mid-June - Corpus Domini. Exactly 60 days after Easter, Catholics celebrate Corpus Domini, which honors the Holy Eucharist. In Venice, this feast day typically includes a long procession in and around Saint Mark's Square; this procession is believed to be the oldest Corpus Domini procession in Italy, dating back to 1317.
Saturday closest to start of summer - Art Night Venezia. To ring in summer, Venice holds a Saturday night of free museum admissions, special events and concerts lasting until midnight or later, similar to White Nights held in other European cities. See Art Night Venezia