Tesla would not want to get into the tire manufacturing business any more then they would want to get into the chip fabrication business. Design is another matter.
I am speculating here... Tesla has already filed a patent for a method of inflating tires dynamically - more or less. Other than electricity, tires are the single most costly consumable on their vehicles.
This tells me they are interested enough in tires to be looking at novel/patent-worthy improvements. Tesla has a history of vertical integration (trying to think if they make their own aero wheel covers?). They may subcontract for a time but in the end they often eat their own dog food and make their own product.
A Tesla tire would not be just another tire. It would need to operate in tunnels at high speeds (maybe not on todays vehicles) and in evacuated tunnels the tire pressure would not be working against atmospheric pressure so a patent to adjusting the air pressure dynamically might work in concert with a Tesla tire of their own design that is particularly tunnel friendly.
I believe tires are a major expense on Semi trucks. Tire improvements are important. I would expect Tesla to be looking at ways to mitigate that expense. Tires are also a significant fleet expense so they almost assuredly looking at tires closely.
Part of what influences Tesla design explorations is Mars. There will need to be tires of some kind on Mars and it is more cost effective to make them on Mars than to import them. And really, the same with chips. I would not be surprised to see them acquire a small chip fab.