On a street car it really shouldn't make much difference. On our circle track car, it does make a difference, but they are going all out around the curves all the time. We send the car out with 25 psi in the right side tires (car turns left, so the heat builds most on the right). With compressed air, after 15 hot laps, on a hot summer day, it may come back in excess of 40 psi. With nitrogen more like 32 to 35. The race tires "grow" more with extra air, and on an oval track car, the rights grow more than the lefts. This causes an increase in stagger (Rights are always taller then lefts to help turn left) which changes the way the car handles, usually for the worse. On street cars, a couple quick turns or burnouts aren't going to have the same effect with heat, and even it they do, the right/left balance is less likely to change.
I have easy access to nitrogen, but it's not worth the effort.