Bathroom exhaust fan heat loss
This isn't typically calculated, but here are a couple of gross estimates of the heat loss.

Method 1:


Air requires in the ballpark of 0.24 BTU per pound to raise it 1 degree F.
1 CF of air weighs about 0.0807 lb, so it takes roughly 0.0194 BTU per cf per degree.
A typical small bathroom fan exhausts about 70 CFM or 4,200 CF per hour.  That would take roughly 81.5 BTU per hour to change it 1 degree.
The exhausted air will be replaced by outside air infiltrating through leakage.  Say you're in an area where the outdoor temperature is 40 degrees colder than indoor temperature in the winter.  That would require roughly 3,260 BTU per hour to make it the same temperature as the indoor air.
If you're heating with electricity, 1 watt = 3.412 BTU/hr, so this example would be roughly equivalent to a load of 1 KW.  Running it continuously would  add about 700 KWH per month to your electric bill in the winter if that temperature difference was an average (roughly $100), not including the electricity to power the fan.


Method 2:


There are online BTU calculators, like this one.  They're used to estimate HVAC requirements, which is slightly different.  In the calculation above, I just looked at the heat (or cooling) required for the replacement air.  HVAC sizing considers maintaining a temperature difference inside insulated walls, HVAC duty cycling, and other factors.  But it's another crude estimation.
For HVAC purposes, turning over room air 8 times per hour is a rough rule of thumb.  So 4,200 CF per hour would represent a 525 CF room.
Using assumptions of "normal" insulation and the same 40 degree temperature difference, the calculator comes up with a 5,840 BTU/Hr HVAC unit.


These are very crude estimates, but you can play with the assumptions to get a ballpark idea of the heating and cooling load created by running the fan continuously.