Just a quick post with a video I made. I borrowed George's geiger counter and thought I'd see just how radioactive a lantern mantle is.
The mantle registers about 4 micro sieverts per hour. This is about 1 half to 1 days worth of radiation depending on where you live. In the US the average annual radiation exposure is about 3600 micro sieverts or 360 millirems. In Canada we get exposed to about half of this at about 1800 micro sieverts per year. This works out to about 9.8 and 4.9 micro sieverts per day for Americans and Canadians. These numbers are just averages and your local radiation count can vary greatly.
In the video you can see the geiger counter has to be almost touching the mantle to get the max reading and at a foot away it isn't reading anymore than background so these mantles are pretty safe. I wouldn't want to tuck one inside my hat and wear it around for a long time or have a stockpile of several thousand under the bed. The real danger is the smoke from these when they are first ashed. If breathed in the smoke can leave thorium and its decay products in your lungs which will sit there slowly irradiating your lung tissue. In turn this could very slightly up your chances of lung cancer.
Newer mantles are made with yttrium instead of thorium because it isn't radioactive.