Will grayscale increase or decrease battery consumption? If I turn on grayscale on my iPhone 6, will the consumption of battery increase or decrease?
It makes no difference on the iPhone 6. So neither a reduction, nor extra consumption.

Some sources that tell the same story:


From a SuperUser question LCD/LED screens; how color affects the power consumption?:


  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.


As the iPhone 6 uses a conventional IPS LCD panel, it means that the power usage is mainly due to the backlight - and the backlight power usage does not change whether the displayed image is full color or monochrome. However it does change according to the brightness setting and how dark the displayed image is.
From CNET's article LED LCD vs. OLED: TV display technologies compared:


  The energy consumption of LCD only varies depending on the backlight setting. The lower the backlight, the lower the power consumption. A basic LED LCD with its backlight set low will draw less power than OLED.