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.
I haven't read any analysis of this, but my guess would be that it would make no significant difference. Really all that it does is average out the colors of all pixels (so red green and blue of each pixel are the same rather than different strengths).
Night Shift does use a little extra battery, though. Apparently it uses more energy to show the yellower hues. But this isn't the same as that.
If you want to use less battery, you can turn on Power Saver mode (are you running iOS 9?). That will save energy by reducing certain things that use more power like email fetching.
Other things to do are to close background apps, go to Settings>Privacy>Location Services and go to the bottom and go to system services and disable things you don't need it to use your location for.
Also wifi, Bluetooth, cellular data, these can all be turned off to save battery. Reduce the screen to minimum as well. Don't do anything on the phone you don't need to. If possible put it in airplane mode or shut it altogether.