The final is always on June 24th. Looking at the calendar this year, the semi-finals should be on June 11 and 12, following what has always happened in the past. On Easter Sunday, the teams are drawn to decide who plays against whom on the semi-finals, we should know more then definitely as whether the matches will go on as they have in the past once the teams are drawn.
The city is not disturbed during the matches, other than what is directly part of the games. There is a parade in historical costume that starts in Piazza Santa Maria Novella - with all of the players included - and they make their way through the center, including through Piazza della Signoria, on their way to Piazza Santa Croce. The parade starts around 4pm, they need to be in Santa Croce by 5 for the game to start but it always starts later. Aside from this outdoor parade passing through some of the main piazzas, most sites (museums, churches, etc) go on as they normally do.