[smoldesu]

I'm not sure. Considering what I know about OLED, I'd assume that #010101 consumes considerably more energy than #000000 since it needs to emit light for that individual diode. The energy efficiency from there will mostly depend on the brightness/size of the display.

[fortnum]

But does it use less power than #FFFFFF?

[smoldesu]

Yes, but it's efficiency scales with the overall brightness of the display. True OLED black will always consume the same amount of power, regardless of brightness.