Nope! I do exactly this. The ATV is effectively a screenless iPad/iPhone (with an older CPU) - it runs the Zwift app natively, and quite well.
You will probably want to use something like an Android smartphone to run the Zwift Companion app as a controller for the game. Not mandatory but it make interaction easier. (typing, changing camera angle etc) If you rarely to never interact with the game during a ride (beyond “riding”) you could skip that, or use the ATV Siri remote included with the ATV.