The concept is doable for sure, but I'd reconsider the draining issue. The volume of a hot tub is quite something, and if you're using any kind of soap or other chemicals, you've got a potential environmental challenge on hand. Even if you only use biodegradable stuff, that doesn't mean that plants will rejoice being smothered with soapy water.
For the heater you'll need something with a decent wattage to get it up to temp in short enough time, so make sure you're not overloading the circuit. Those outlets in some shed are often not the best.
Propane/butane would work as well, but in that case you'll need to take care of proper ventilation as well. Or just leave the door open.