Border crossing and rush hour traffic aside, it's less than a 2 hour drive from Burlington to Montreal by highway. I do that trip occasionally on my bicycle from north of Burlington and it's only 75 miles (on back roads). There isn't much to see on the way by taking the highway, but you could take Route 2 through the Champlain Islands to the border, and cut across the Quebec farmlands after that. If you're interested, I could give you a route using backroads that will take you across the border at a small crossing without the traffic backups of the larger ones, and can send you down a road that had one of the most spectacular home flower gardens that I have ever seen.

If you're staying in downtown Montreal, you'll probably find it hard to escape from paying for parking, unless you want to shuffle your car from space to space. However, if you're going to be there for 5 days, you may want to have a car for part of that time to see things away from the city (Laurentian mountains, etc.). You may be perfectly happy spending 5 days right in the city, it all depends on what you like to see and do and the pace at which you like to do it.

You didn't ask for any restaurant suggestions, but let me mention Le P'tit Plateau, a very good bring-your-own-wine French bistro that is not at all stuffy and is reasonably priced.