Do larger rims and smaller profile tires give you better gas mileage?
Yes and no. It’s not the tires and rims that give you better gas mileage. You may get better gas mileage because lower profile tires generally require higher pressure. The higher the tire pressure, the better gas mileage you’ll get because there’s less friction between the tires and the road. Imagine trying to push yor car with 4 flat tires. The friction would make it impossible without help. You can get the same gas mileage benefits by overinflating your tires. But then they’d wear out in the middle of the tires prematurely and you’d have to spend money on new tires.
It is not the wheel diameter but the outside diameter of the tyre that counts. If you have a lower profile tyre on a larger diameter wheel you may end up with the same outside diameter. This outer diameter does affect the speed and fuel consumption and also the speedometer, which can sometimes be re-calibrated.
Julian M got it correct. It is called the rolling radius. I.E. the outer circumference of the wheel/tyre should be the same as original. Therefore for 1 revolution of the wheel it would travel exactly the same distance over the ground. So without any other mods it does absolutely nothing to mileage or performance.
Just to add to SteveO’s explination. Larger rims wouldn’t be necessarily be heavier, they’d just be harder harder to spin as you would have more weight further from the center. I could get all technical and say that the torque needed to rotate a tire isn’t much, but it really adds up once you consider stop and go traffic and constant acceleration. Lower profile tires w/ bigger rims are normally for higher performance as they offer better sidewalls which in turn mean better handling. For better mileage you’re better off going the other way and getting a smaller light rim and offsetting it with a bigger tire. People don’t really do this though because it looks funny.