Hi,
Certainly a "Directly Driven" LCD display (for example as used in a wristwatch) can have extremely low power consumption. But the "serial" displays use a Multiplexed Display Controller (such as the "HD44780") which the data sheet implies requires about 0.5 mA. That's similar to a PICaxe running at 4 MHz, so the best you might expect is a total power consumption of 1 mA, with no backlight (or negligible current in the OLED elements).
It rather depends if you consider 1 mA to be "low power" consumption. It is, compared with a "normal" LED backlight, or many other components such as a radio receiver, relay, voltage regulator ("7805", etc) or even poorly conceived switches or drivers, etc.. But it might be excessive for "24/7" operation from a small battery (AAA or AA, etc.).
One of my present "projects" is using a PICaxe to drive one of the "I2C expander backpacks" on a backlit LCD display and I've just measured the "power budget" as follows: The PICaxe (08M2) draws about 1 mA (because it's running at 16 MHz) and the total battery drain rises to ~5 mA when the LCD module is connected. That's more than I would have expected; I think much might be due to the red "pilot light" on the back of the I2C backpack which performs NO functional purpose! The (green) backlight is controlled by PWM and the "dimmest" settings (still adequate in indoor room lighting) add almost nothing to the current consumed (~ 100 uA).
In summary, I suggest that for a "low" current application, there may be other factors than the difference between LCD and OLED technology. However, it may be (much) easier to lower the LCD backlight current (e.g. by not connecting it! ) than an OLED. But I don't currently have a hardware setup to do a "side-by-side" comparison. The other significant factor is that a backlight consumes the same current whether any of the LCD pixels are "lit" or not, whilst the OLED current rises when more pixels are (actually) lit.
Cheers, Alan.