In general, on a turbo'd car, having an open airbox results in higher air temps, as you're letting the heat in the engine bay more directly affect the air going into your air intake. The reason it matters whether or not you're turbo'd is just because turbocharged cars tend to run warmer than NA cars.
If you have a good intercooler, the effects can be somewhat negated, but an intercooler only cools air that is pulled through the turbo, meaning your air coming through your intake and not running to the turbo is unaffected by the intercooler. So at the end of the day, you're still getting some hotter air into your engine, which results in worse performance.
I'm oversimplifying a lot here to make the explanation simple, but that's the general rule of thumb. It can vary greatly car to car, depending on engine position in relation to the air intake, how effective your cars cooling systems are, how good your intercooler is and even what the weather outside is like. Muggy hot days will result in worse air.