When a color LCD/LED display is off it is black, when on it is colored. What color to light is an LCD/LED display When a modern display using pixels is turned on its colors are different than the black that the screen is actually made up of. When put in light, such as sunlight, does the screen react to the color that the screen is currently displaying or to the actual black pixels it's made up of. That is to say does it heat up as a uniform black object, or if the screen is  displaying white will it heat up slower reflecting the fact that the screen is displaying white (or any other color for that matter). 
 A: LCD displays work by absorbing light, which is why they need a backlight (or a reflective surface) behind them. I think their default state is to not absorb light i.e. to make the screen black requires power, though I wouldn't swear to this, so when you're looking at an LCD display that is off, you're looking straight through the liquid crystals and seeing whatever colour the backlight is when it's off.
I'm not sure what you're asking re the behaviour in sunlight. When powered off or set to white the liquid crystal doesn't absorb light, so the liquid crystal doesn't heat up but whatever is behind it will heat up. When the LCD is on and set to black it absorbs light, so the liquid crystals will heat up but the screen behind them won't.
LED screens are completely different. They use light emitting diodes to generate light. I don't think light absorption by a light emitting diode greatly affected by whether it's on or not, so they will heat up at roughly the same rate whether they are on or off. Note however that LEDs generate heat, when powered on, due to resistive heating and unless they're in unusually strong light the majority of the temperature rise will be from resistive heating.
