Q: How can I stop messages (texts) from displaying on my tv while using airplay from my iPhone to apple tv?

If you use mirroring you can't, since that's what mirroring is, it mirrors your screen. If the content you are watching supports it try turning mirroring off and just use basic airplay.

Definitely prefer using basic AirPlay over mirroring, but if you cannot, setting do not disturb might work. It's the moon in the Control Center.