You're asking people who think that putting heat into castor oil keeps their engines cool?
Looks like Steve found some thermo stuff on the World Wide Web. Sure, Steve.
I do things around the house all the time based on sound thermodynamics, and my wife (a mechanical engineer) still yells at me about them. My first thought is whether letting the water stay in the tub makes it harder to clean. Next consideration would be animals. Little kids--not allowed in my house-- could drown. Large, woolly, water dogs, e.g. the late Chumley, might gleefully immerse themselves in the water, then shake it off in the kitchen, exacerbating the evaporation problem that so concerns Chuck.
So let's get quantitative. Help me out here, Steve. Google says the heat capacity of water is 4.18 J/g*C. I guess C is degrees Celsius, and it's in the denominator. Alexa tells me that the standard bathtub holds from 35 to 50 gallons of water. Call it 35, assuming that the occupant was equivalent to much of the other 15 gallons. 35 gallons weighs 35 gallons* 3784 grams / gallon = 132,440 grams. Alexa says 40C water is hot. To what temperature would it cool? Temperature would decay in pretty much e
-kΔT, but what the heck is k, and what's the low temperature? Use a thermometer to track water temperature. If you have a setback thermostat, which you would if you are this concerned with energy recovery, check the temperature somewhere around dawn (or maybe next Thursday if you're in Texas). I'll guess 65F = 18.33C. So you'd recover 4.18 * 132,440 * (40 - 18.33) joules = 11,852,996 joules. Alexa says 3.6 million joules = 1 kW-hour, so heat recovered = (11,852,996 / 3,600,000) kW-hr = 3.292 kW-hr. My marginal cost of electricity is 11 cents/kW-hour, so you'd recover 36 cents worth of electricity. Electric heat is 100% efficient, so that'd be worth 36 cents if you have electric heat. Gas would be more complicated. It would depend on your furnace efficiency. If you have a fancy geothermal heat pump like JCT Manor, it's even more complicated, but 36 cents is maybe an upper bound.
Heat only flows from hot to cold. The bath will never reach thermal equilibrium with the house unless the air in the house is at or below the dew point.
The hot bath will fall to the same temperature as the house, but as long as the water is evaporating from the tub it will take heat from it's surroundings. The heat of evaporation is lost until the water vapor condenses, at which time it is released. The heat required to make the phase change from liquid to gas is rather high for water.
Folks decided above that the humidity is good, but Chuck points out that it's not free. One could avoid evaporation by putting a sheet of Saran wrap or equivalent on the water surface (Marilou and I wonder why California doesn't do this with its aqueducts) or by pouring used motor oil on the water. One probably changes his oil less frequently than he bathes, so a combination of the two techniques might be indicated. I muse that the ratio of oil changing to bathing is possibly invariant with socioeconomic status within a certain range.
Speaking of electric heat, my favorite thermodynamic thing is telling people who have electric heat that they can leave their lights on in the winter with impunity.