I understand your answer that it is better not to use the hot water on Shabbat because you will be warming up cold water. However, in my parents apartment building cold water only enters the water heating system when the water level goes down to a certain point. Therfore, if I used the hot water, cold water may not replenish the hot water that I just used. In a case like this is it allowed?
I think this is a case of what is called Grama i.e. an action that causes a prohibited Melacha in an indirect way. This is not a Torah prohibition but still a rabbinic one. I therefore think this system should not be used on Shabbat.