Visiting Montreal, do I need to drive?

Montreal has an excellent and safe public transit system.

They have the subway (called the Metro) system that pretty much goes all around the central core of the island of Montreal, and even has one line (Yellow) that goes to the south shore, and one (Orange) that goes into Laval to the north. The only places it does not reach is the West Island area, which is serviced by trains and buses. The only problem with the subway is that it shuts down around midnight and only starts again about 6AM. So if you are out partying until 3AM when the bars and clubs close, you'll probably be walking or taking a taxi back to the hotel.
http://www.stm.info/en/info/networks/metro

There is also various bus routes both during the day and overnight. The trains run from about 6AM to midnight.
http://www.stm.info/en/info/networks/maps

If I were you, I would suggest you avoid getting a rental car UNLESS you actually plan to leave the island of Montreal. If you plan to head out to the Eastern Townships or up north to the Laurentians, then you will need a rental to get there, since public transit is limited once you get out of the metropolitan area. But on the island, you can reach all the tourist areas (Old Port, Downtown, museums, casino, Plateau district...) quite easily. Not only does a rental car cost you $50-100 per day for the vehicle/gas/insurance, but you also have to pay to park it in most parking lots in the downtown area and at places like the Olympic Stadium. Hotels will generally charge $10 or more a day for parking at their lot. And gas just hit $1.52 per litre yesterday (about $6 per gallon).

So no car. Just walk around. When it rains, use the tunnels for the Underground City in the downtown core (a series of malls and buildings connected by underground pedestrian walkways). Use the subway to get between downtown and the Old Port area. Take a horse-drawn carriage ride or rent a bike in the old port area. Maybe get a BIXI card and use the public bicycle stations around the city.
http://montreal.bixi.com/

Absolutely not!!! The core of Montreal is accessible by public transit. The metro runs underground and is really clean.
6$ for day pass for bus and metro.

if you can utilize the Metro, please do! It's much easier than trying to find parking. You can get a variety of passes - 1 day, 3 day, 1 week

We have excellent public transport.

Also the transit comes every ten minutes. You never wait long. Don't forget transfers.

Using google maps, see if you can get where you need to by transit. The Metro is quite good.