The only place in Burlington where it is illegal to ride your bike on the sidewalk is the "Inner Fire District", which includes the blocks immediately surrounding the Church Street pedestrian way. Of course, when you ride on the sidewalk you should behave as a pedestrian and give way to more vulnerable users. Also watch out for curb cuts because that is where motor vehicles enter and exit across your path.
I think it is much safer for all to ride a bike on the sidewalk. Needless to say people walking have the right away. but it is easier for a bike rider to give the right away than it is for an operator of a car to give the right away to a bike rider. And we would not have to put up with a lot of money being spent with all the traffic control now being done for bike riders....
I believe you are supposed to behave as if you were a motor vehicle, not sure how you can do that on a sidewalk. It is really unsafe to ride on the sidewalks..... the most important thing I would point out is that cars are not expecting fast moving objects to cross intersections and driveways.
As Ben Westermann-Clark notes riding on the sidewalk is technically legal in most of Burlington and there is a particular etiquette one should observe. However riding on the sidewalk greatly increases your crash risk. It is also legal to practice "lane control" pretty much everywhere in the City (including the "Inner Fire District", but obviously not on limited access highways, e.g., Route 127 and the Interstate, where bicycles are banned) and such a practice will generally be safer than sidewalk riding, although it may take some getting used to. Hopefully in the coming years the City builds a safe, comprehensive network of protected bike lanes that makes both these strategies needless.