Is the transport of water molecules an active or passive process?

In biological systems, movement of water is ALWAYS passive. Water diffuses from regions of high water activity to low activity. Such movement (osmosis) may be in response to the active transport of other molecules (e.g. sodium), BUT water itself moves passively--from "high" to "low." Always.

Someone might try to get cute and claim that since contraction of the heart requires energy, that would mean that the pumping of blood involves "active" transport of water, but that's just playing pointless games--movement of the blood is just a passive response to the creation of a pressure gradient by the cardiac pump.

Depends.