Racers usually pay attention to tire diameter when they are running foam tires. Foam tires are typically used in on-road applications and sometimes dirt oval.
In those cases, racers will often true their tires, sanding them down to a particular tire diameter they are looking for. Top-level drivers will often leave very little foam on the rim, ensuring that the wheels have as little mass as possible, making the car faster and more responsive. You will usually see a section for tire diameter on setup sheets, so you can see exactly how far racers are sanding down their tires.
Be aware, if you true the tires down so they're close to the rim, their lifetime will be much shorter, and you will need to adjust the ride height of your car to compensate.
Tire diameter also affects the overall drive ratio of the car. Large tires will behave like using a large pinion gear - you will have less acceleration and more top speed, and conversely, smaller tires give more acceleration and less top speed. Because of this, use the car's rollout to determine your gearing, as relying on just the pinion and spur is not enough.