This depends on the display technology being used. Most LCD devices use a backlight which is always on. When an LCD is black, it is doing full light output, and then also blocking all of that light to create the appearence of black. (This might make you ask "Why?", but remember that the backlight does not have as fine-grained control as the pixels themselves. If you need to make just a few black pixels, you can't just turn off the backlight in that area, so the backlight is always on and we just block light for those few pixels)
There are some technologies that aim to reduce power when large sections of the screen are black (you might have heard of "localized dimming" or "dynamic contrast"), but these features are largely found in TVs where movies might actually have large swaths of black on screen. Computers generally don't.
(In order from most power-consuming to least power-consuming)
- CRT technology works without a backlight, but uses a high-energy electron beam to make phosphors glow, which means that a black background will significantly reduce power consumption.
- LCDs with CCFL and LED backlights will have little to any power change based on if the screen is white or black because the backlight is always on (an LED backlight will use less power than a CCFL backlight, however). A white or black background will use less power depending on the resting state of the LCD. If it blocks light by default, and requires power to unblock the light, then a black background is better. If it doesn't block light by default, and requires power to block light, then a white background uses less power.
- OLED technology works without a backlight, and a black background would reduce power consumption.
If you want to reduce power consumption, configure your system to sleep the display after 10 or so minutes, as this uses the least amount of power and will allow LCD displays to turn the backlight fully off. Generally when you're using the computer, you're not staring at the desktop wallpaper, so the biggest gains are available when the system is not currently in use. Turning down the brightness will also provide significant power savings. I generally run my displays at about 20% Brightness, and get more than double the battery life in many cases. This won't work if you're out in the sun, but it's often plenty bright for most indoor use.
For desktops, you can also consider getting a display with a different technology that consumes less power - a CRT uses roughly 5 times as much energy as an LCD with CCFL. An LCD with a CCFL may use as much as twice the energy that an LED backlight does. I'm having trouble finding numbers for full-sized OLED monitors, but if I remember correctly OLED technology is several times more power efficient than LED-backlit LCDs.