I think it's safe to say many, many people don't do this.
If you have a water dispenser (through a refrigerator), do you fill and dump two glasses before getting yourself a glass of water to consume?
I guess I would. It seems dangerous not to for the reasons mentioned here.
I hadn't got a water dispenser like that but I'd be really surprised to find that the internal reservoir and piping was made of anything except for plastic.
Some actually have coiled copper (it transmits "cold" better) - but almost all of them have a filter.