A wood-fired hot tub harkens back to the days before electricity or access to natural gas fuel. Many hot tub purists like the natural experience of soaking in a spa that is heated by burning wood. If this appeals to you, you can still get that kind of hot tub from several manufacturers.
Using a wood-fired hot tub does require a bit more work than electric or gas hot tub heaters, but the basic principles are the same. Water from the tub reservoir passes through the heating element, then through a filtration system, and finally back into the spa.
The essential difference (from a water-heating standpoint) is that a wood burning hot tub heater cannot keep your hot tub's water at a precise temperature, the way the other heating options can. Moreover, the heat will tend to stratify, meaning that the hotter water will rise to the top, while the cooler water sinks to the bottom. To combat this, use a paddle or similar implement to stir the water occasionally while you are soaking.
The basic operation of a wood-fired hot tub focuses on the heating stove, maintaining the proper water level to cool the heater, and the airflow. When starting a fire in the stove, it's best to use clean-burning paper for the initial ignition, followed by dry kindling, overlaid with criss-crossed logs. The rule of thumb here is simple: the more wood you use, the faster the water in your hot tub will reach soaking temperatures (about 104 F). The more wood you add, the longer the water will remain at the proper temperature. If the water gets too hot, damp the fire and reduce the airflow into the stove, both of which will cool things down pretty quickly.
Take special care to keep the wood heater's exterior properly cooled. This often means water-cooling, and it requires surveillance in order to maintain it. Some stoves are made of metal (aluminum, for example) that will melt if you fail to monitor the cooling system.
Smoke is the other potential problem with a wood burning spa heater. Using a high-heat, low-smoke fuel source helps reduce the amount of smoke, as does making sure the wood is very dry and not green. Keep the stove's chimney (if it has one) clean and obstruction-free. This is very important!