Air filled tires are recommended to fill to 32 psi. During use, they heat up and pressure increases.
I read, somewhere, that pressure in an air filled tire increases by 4 to 5 psi during usage.

If this is the case, the nitrogen filled tires never reach operating pressure so... my question is, should nitrogen be filled to a higher pressure ... the operating pressure of air? If not, you never run nitrogen and full operation pressure (as compared to air).

 

It depends.

In your normal "air" filled tire, you're already at about 78% Nitrogen anyway. The other 25% is made up of other gasses that occur naturally in the environment - everything from oxygen from plants, methane from farts, to microorganisms like yeast. There's also water vapor that's making up a couple percentage points.

So assuming that nitrogen doesn't increase pressure when it's heated, that means the other 22% of free radicals in your "air" tire are the culprit, and you'd want to get rid of those.

So you have your tire guy put on the new tire (which has ambient "air" from the shop sitting in that unmounted tire), then he puts on a fancy green valve stem, hooks the Nitrogen hose to it and POP! the tire is mounted. But there's still that ambient "air" in there (because it wasn't purged out before introducing the Nitrogen), so it'll still cause the tire pressure to change when it's heated.

Or maybe you got fancy, and bought an aftermarket wheel that has TWO valve stems, which will enable the air in the tire to be purged out and have the environment in there be pretty close to 100% Nitrogen, in which case you probably know the sweet spot that your car and tires like anyway.

So why did I start this all of by saying, "It depends"? Because every tire is different, and only you will be able to determine what the proper warm temperature should be. If you want to cut your research time, get yourself a pyrometer and take a reading across the tires' treads to see if they're being used efficiently when warmed up.

Long story short, unless you're planning to crack a speed record at Bonneville or turn hot laps in your Formula 1 car, whatever gas you put in the tire should probably follow the recommendation of the manufacturer.

 

The bottom line is that nitrogen expands with temperature the same as air does.

At the conditions found inside a tire, either would be considered an "ideal gas" and change pressure according to Boyle's and Charles' laws. Filling tires with nitrogen has the supposed benefit of removing oxygen and water, which could slowly degrade the rubber and corrode the rim.

Tires are designed with a normal pressure increase in mind and specified pressures are always when cold. Don't let air out of a warm tire because you think the pressure is too high. Wait until it is cold and measure then.