I tried this once on a cold winter day: I let the bath cool down to room temperature (which was <20C) before letting it drain, with the bathroom door open. The result was a lot of condensation on the windows in the surrounding rooms. I figured the high humidity was going to cause more damage than it saved money, so I scrapped the idea.
Although higher humidity should make it feel warmer given the same ambient temperature. So if you're increasing the temperature mildly using this method AND increasing humidity, it may compound one another positively. Although, it is a minor efficiency gain either way.
At the moment, I've got two beefy dehumidifiers running and I'm still at between 70-85% humidity in each room. I don't have any damp or leaks, it's just that the outdoor humidity has been 80-90% for the last few weeks, so ventilation is futile. And because of the weather at the moment, we're drying clothes indoors.