There are wood hot tubs, and there are wood burning heaters. I've worked on a few wood tubs, some were still slats, and they leak if they dry out but seal up once the wood swells, and one with a liner. I've only seen wood burning heaters on tv and in ads.
You must use chemicals in any hot tub. Which you use can vary, but they all need sanitizer and ph adjusters. I have no idea how they maintain a wood burning spa.