While the Earth is (mostly[1]) a closed system when it comes to water, any particular region on Earth is not a closed system with respect to clean drinking water.
Deplete groundwater reserves faster than they refill? No more water for you. Raise the ambient temperature far enough that the rainclouds go somewhere cooler? No more water for you. As long as your usage is below the replacement rate, all seems well. Imbalance the system, and you have indeed "consumed" your local water supply.
> Is the water contaminated with chemicals? Maybe heat?
Typically yes. Definitely yes.
> Can people or wildlife consume it?
With enough water purification infrastructure, sure. I'm not aware of very many datacenters installing such infrastructure.
> Can you use rainwater?
It's all rainwater, whether directly or indirectly. The question is what would have benefited from that rainwater had it not been redirected to a datacenter?