In practice feed vendors who sold grain to tourists wanting to feed the birds have been banned, and other moves to shoo the birds away have been taken by the city council, in the face of protests from animal rights groups. As a result, the pigeon population of St Mark's Square has been reduced from an estimated high of 20,000 to barely a thousand. Read here (and here) to learn more.
P.S. From an anonymous source, close to the city council, I have learned that from now on the official pigeon-related quote the lagoon city will adopt could be the following, by George Bernard Shaw,
Remember that you are a human being with a soul and the divine gift of articulate speech: that your native language is the language of Shakespeare and Milton and The Bible; and don’t sit there crooning like a bilious pigeon.