LCDs can be made either way, with the "natural" state being either the open or closed position. Simple LCDs like you'd see on a digital watch or non-backlit alarm clock, where the "background" is that sort of greenish grey color and the digits are black, that is an LCD with a natural "open" state. Monitors and other computer-type displays use LCDs with a natural black state instead. As far as I know, this has always been the case, but maybe really old (primitive :3) LCDs used "open" LCD panels instead. Either way, whether an LCD panel uses more power for white or black, it is a trivial matter. The majority of the power used by an LCD display assembly is consume by the illumination source, the panel itself uses miniscule amounts of power.
For CRTs, it may be true that they use less power to display black, I'm not sure, though it is something that can be easily measured with a wall meter. Unfortunately I just got rid of my last CRT a month ago or so In any case, even if it's less, it's not going to use "no power" or even close to that to display black. Like LCDs, CRTs have multiple parts that work together to operate the screen; in this case it fires a beam of electrons through a vaccuum tube. The beam makes the phosphors on the screen glow. The beam itself fires in a fixed direction, but they use electromagnets to alter the trajectory of the beam, in a systematic fashion over time so that it "scans" from left to right and top to bottom. This points the beam to the correct pixel, and at the same time, the intensity of the beam is modified at just the right moment to light each pixel to the correct brightness as the beam passes by.
So anyway... while you could say that when a pixel is black, the electron beam intensity can be zero, which takes less energy. But that is only one component of the screen; the display unit as a whole will still use power, because the electromagnets which control the trajectory of the beam and everything else are still operating. How much less power it takes when displaying black vs white depends on the ratio of power that is consumed by the magnet system vs the electron beam, and I have no idea what that is. Would be interesting to measure though.