Many data centers use evaporative cooling, since it's much cheaper than closed loop. This requires a constant input of water.

Yes and just like our bodies, that closed loop is cooled by a rack of evaporators on the roof.

The blood comes from liquids I ingest (and some food and air). If I don't drink, I'll die quite quickly.

That isn't a closed loop exactly although there is a complex system connecting my digestive/urinary tract with my bladder etc.

If this is true, why have I heard about data centers demolishing local water supplies? Is this incorrect?

Apparently, you don't urinate like the rest of us; humans are not a closed loop. Neither are most data center cooling facilities, but I think it's pretty clear that level of fact is wasted here.