Not very. The predominant isotope of thorium is an alpha emitter with a half-life measured in tens of billions of years.

Looking at the video, the counter appears to peak around 200cpm, or (per the Grafana dashboard screenshot) about 2 microsieverts per hour. That'd take 50 hours, a little over two days, of exposure to add up to the dose of a typical chest X-ray. And that's for a whole mantle, placed directly adjacent to the tube; it's not obvious in the video, but the dose rate is highly dependent on distance. Spicy gamma emitters aside, most small samples drop to background just an inch or two from the detector.

So I wouldn't worry all that much about it, really. Eating a mantle, or breathing the dust from a freshly replaced one, wouldn't be a great idea; I'd expect it to increase cancer risk to a degree probably commensurable with some number of years of smoking. (Probably fewer years than I racked up, back when I still had the habit.) Nor would keeping a mantle in your jockeys, or your bra. So don't do any of those things.

But buried, or even just mixed with dirt? I'd worry more about falling into the fire, and that's not something I worry about a lot, even if it does typically get pretty drunk out every time I ever go camping with anyone.

For comparison I have seen statistics that a granite countertop, or granite flooring can emit around 1.7 microsieverts per hour. Granite is mildly radioactive. I imagine working in an old 1920s era office building with granite column facing materials and granite hallway flooring would be a lot more radiation load on the body, unless you spent dedicated time every day seeking out and handling gas lamp mantles.

Lots of people have marble and granite countertops in their kitchen...

They are quite radioactive for a household item indeed. The dust is a good point I keep that one double bagged for that reason, although they aren't as fragile when they're new.