...its water's heat into your bathroom! 

When you have a bath in colder months, leave the warm water in the tub and notice how warm the bathroom stays for a long while thereafter.  The heat in the water (which you paid good money and used good energy to heat!) radiates into the room.  If you pull the drain too soon, that heat will escape your house along with the water and heat up your sewer lines instead! 

Let us not spend all this money and energy to heat water only to use some of its heat to soothe and cleanse us but waste most of the heat energy stored in the water by pulling the drain-plug.

Get full efficiency out of that heated water by leaving it in the tub until it's at room temperature.  If you wash dishes by hand do the same with the used dish-water.

How does this work?  Water is a very efficient 'thermal mass' which means it stores heat, like a battery stores electricity, and releases it when the ambient air around it is cooler than itself.