Do wheel types affect driving?

Original Answer

You're not effecting the tire choice of the car, just the wheel rim.

I've tried many different wheel and surface combinations in a number of vehicles and haven't noticed any difference in terms of the cars handling.

Someone may be able to disprove me with numbers or something more empirical, as my evidence is purely anecdotal, but aside from the bulletproof tires, I believe all of the wheel mods are purely cosmetic.

EDIT: I've revisited my answer below.

Updated Answer

Some people with more time and patience than myself have taken the time to run a few tests and document everything.

Like this reddit thread where the poster records his times using the same vehicle with various upgrades on the same track surface, including stock car with each wheel type. This particular set of numbers reflects an extremely negligible difference between the wheel types.

There is also a YouTube video showing a slight difference in traction for the Off Road tires.

There is also this reddit thread where the poster comes to the conclusion that the difference is somewhat negligible.

So, there is a lot of anecdotal evidence and some video, but I don't think anyone has found any confirmation in the code that wheel type has any modifier effects for a vehicle.

In summation, your mileage may vary.