Welcome to America. You should know our federal and state warranty statutes say that your warranty stays in effect until and unless the warrantor (Audi) can actually PROVE that what you did, broke their car. So putting on different wheels and tires, which are replaceable items, cannot void your warranty unless Audi can prove they damaged the car somehow.
The TPMS system uses the ABS wheel-rotation-speed sensors, as long as they can sense your wheels are turning, that should be unaffected. I'd expect there's some point at which a third-party wheel might not be seen properly by the sensors, which could cause all sorts of problems, but the wheel manufacturer should know about that.
One thing that probably will be affected is your speedometer and any function ties into that. Unless the diameter of the mounted wheel and tire is exactly the same as it was on the original? The speedometer will need to be recalibrated. That happens on all cars, if your wheel change also means an overall diameter change. If you drop to lower profile tires to compensate, that can work, but that's also going to change the ride and handling of the car. Assuming you can drop far enough to compensate.
Since Audi makes 20" wheels, it shouldn't matter that you use third-party parts, they should be able to do the speedo calibration although I'd bet that means Programming for a fee.
It followed me home. That's my story and I'm sticking to it.