You own link points out that the difference between humid and dry air in terms of heat capacity is a low single digit percent. Unless you can reduce the humidity from "a lot" to "basically nothing" and lose only a couple of degrees in the process (highly unlikely, completely impossible if your method consists of opening windows/doors in the winter) it is going to cost more energy to vent out the humidity and heat dry air than it is to just heat the humid air that's already near temp.

Just because a rental contract includes a clause doesn't make the intentions of that clause consistent with reality.