Do I need FAA permission to install tundra tires on a Piper PA-28 Cherokee 140?
Any time you make a modification to a certificated aircraft you need FAA approval. Even what appear to be minor things, like replacing you sun visors with better ones requires an STC. Putting in shoulder harnesses requires and STC. I could go on forever, but you get the idea.

Putting tundra tires on your Cherokee is a major change. If you look in the parts manual, your Cherokee probably came with 4 ply 6.00x6 tires so to change them you need an STC. You would also need to get an STC for the wheels and brakes. You’d probably need a new nosewheel fork as well.

I have never heard of anyone putting tundra tires on a Cherokee, but stranger things have happened.