It might help to think of the tub of water as a heat battery/capacitor: the energy you put in is the energy you get out. So, yes, it will slowly absorb energy when you are heating the room. However, it will also release an equal amount of energy as the room cools down. On that front I'd say it equals out.

Technically the tub of water will change how fast the room changes temperature. Because the difference between the room temperature and outside temperature affects the rate that energy escapes the room, this could affect how much energy the room emits to the outside. For example, imagine that you had so much water that the room only went down to 65 degrees F throughout the day; that means that you put enough energy into the bath battery to heat the room to 65 degrees F throughout the day.

The other way that you would be consuming extra energy is by evaporating the water. At <70 degrees F and little air circulation, I don't imagine that evaporation is consuming a lot of energy. A bath tub of room temperature water doesn't seem like it would be very effective as a swamp cooler.

My guess is that a bathtub of room temperature water wouldn't measurable affect your heating bill.