Does a bad master cylinder cause brakes to stay on?

Not usually. It is possible for the mater cylinder get suck in the apply position, but the brake pedal would not return to the up position after stopping. A more common problem for the brakes to be stuck applied is a locked wheel cylinder or brake caliper, or a jambed parking brake. These issues will be at one or more of the wheels, so check there if your brake pedal is at normal height.