The short answer is no.
The process you have heard about helps to concentrate milk to make it less bulky to transport. Most of New Zealand milk is turned into milk poweder eventually. So no extra water is being removed, it is just getting removed earlier in the journey.
Onto your second point.
For every 1L of milk a cow produces, that cow has to drink 2L of water for starters. That cow also has to eat a significant amount of grass, that grass requires a large amount of irrigation. I do not know the exact figures but to produce 1L of milk might require 10L of water.
So even if the farmer gets all the water back from the milk it does not provide enough water to make a significant impact on irrigation demand.