If I don't treat amblyopia, will my eye shut down with time?

More or less. The 2 cameras (eyes) that provide video input are used by the brain as long as the image quality is acceptable.If one provides poor clarity or is offset enough to provide double vision, the brain will ignore the image.Over time, the ability to receive/use or reactivate the image from the bad eye declines. If you want to lose what vision you now have in that eye, forgo treatment.
Not completely. If you don't treat amblyopia in a young eye, the eye will be less likely to see well over time. It may not shut down more over time, but your brain will not use it and it will not improve. Your best option is to follow your eye doctor's advice about treatment.
Not really. Amblyopia, if not treated will lead to a certain final subpar vision which, depending on the cause of the amblyopia can be 20/25 or hand motion vision, but not 20/20 by definition. This is not to say that some other condition (cataract, glaucoma etc) can't make it worse still. If you see a deterioration after a long period of stable vision, get it checked by your eye md.