I wish to get fit, therefore should I have a heavy bike?
I'm going to assume that your definition of "get fit" is to lose weight and develop aerobic fitness. I'm also going to assume that the person in question is rather unfit to begin with.

All in all, a person can get fit (or not) on either a heavy or light bike; it will really depend on how they go about their fitness program. 

To achieve aerobic or cardiovascular fitness one needs to engage in an activity that elevates the heart rate to a target zone and keep it there for periods of time. To lose weight, one must burn more energy (calories) than one consumes. In order to do this the activity needs to be done on a regular basis. 

On to the weight of the bike... Say a person can choose either a 15lb road bike, a 25lb mountain bike, or a 35lb commuter bike for their fitness program. It makes absolutely no difference which bike they choose in order to lose weight and develop aerobic fitness. The only thing that matters will be how they use the bike. So to get fit, the choice of bike would depend more on the type of use a person desires rather than the weight of the bike.

There are also drawbacks to choosing a heavier bike for the sake of it being heavier. One is that it would be harder on the knees of an unfit person, and even a fit person, if they have to climb hills. Another is that it's just plain harder to have to lug around. And also, it would be much less enjoyable once the person gets fit.

My second job is working as a fitness trainer. If a client came to me wanting to get fit with cycling, I would not recommend that they get a 'heavy' bike. I would recommend that they get the best bike they can reasonably afford, based on the type of riding that they prefer to do. I would also probably recommend that they start out with steady, regular riding and build up to interval training. The question is about getting a heavy bike for fitness; and I'd say no, don't get a heavy bike for fitness. Get a bike that will be frequently ridden.

As a trainer I've had several clients who turned to cycling for fitness reasons. One of them got an 'urban' bike decked out with fenders, rear rack, panniers, etc and proceeded to lose 30lbs and otherwise improve her fitness with a daily 16 mile RT commute, plus weekend fun rides. Another client went with an expensive, light road bike; he managed to burn off 25lbs, and 6 months later rode his first century. So, the weight of the bike shouldn't be the determining factor. Like others have said, it should be about the fun and enjoyment and finding a style of riding that'll be sustained.