It depends.
If the overall tire diameter is identical, and the wheels don't weigh any more than the old wheels (possible, but unlikely unless you're going high-end), then you won't have to replace anything.
You might wear out your suspension faster, though, because the shorter sidewall on the low-profile tires will absorb less impact than the taller sidewall on the 16 inch normal-profile tires did. Other than that, there should be little change.
Bigger wheels don't mean you HAVE TO replace your brakes, it means you CAN replace your brakes. It gives you more clearance so you can put on bigger brakes.
Now if you go with larger wheels with a larger overall tire diameter, it gets more complicated. Your speedometer will read low, because it takes speed from spinning parts of your drivetrain. Say you got wheels with twice the stock circumference, your speedometer would read half as fast as you're actually going. You'd need to recalibrate that.
These would probably be heavier, so you might want to get a tougher suspension.
Also, if you have to raise or lower the car, your camber might be off, so that would have to be adjusted, or else you'd get uneven tire wear and less traction.
You'd have to make sure you clear the fenders. If not, you might need to get them cut to accomodate the wheels, or have the wheel wells modified so they can go up inside. You might end up needing flares installed.
So...maybe not, but probably yes.