Conventional LCDs and LED based LCDs consume power that largely depends only on the intensity of backlight. This means to save power you want to reduce brightness of the screen.
OLED displays, like those found in Samsung (and other) smartphones, on the other hand, have power consumption that depends on color being displayed. If i remember correctly, it was something like
Red<Blue<Green, but that order would be device dependant, I remember that some smartphone had it as
R<G<B. Basically in such situations, white is your enemy, search for apps that have night mode, since black color consumes minimum(not 0) power.
However, to answer your question, since AFAIK OLEDs havent gone mainstream beyond phones as of yet, Darker backgrounds will not save power, but darker display over all (Reduce brightness) will.
Edit: Dynamic Brightness, as mentioned by @Bob, might be used in LED based LCDs which might reduce power based on darkness of color.
A way to test this is to display a black screen in a dark room and see how close to black it actually is. Typically it would be very bright since the backlight is still blazing away even if the LCD tries to block most of that light.