I can understand the worries about nylon, and my personal choice is cotton or polycotton. However, there are quite a few people who use wood burning stoves in nylon teepees.
Look for info on Helsport tents and stoves for some inspiration.
It would need some thinking about though. Not to be done lightly and without understanding all the factors involved.
For example:
Flue height compared with tent angle – sparks/embers come out of the end. Is there time for it to cool before it’s likely to hit your tent.
Temperature of the flue – my titanium one glows red at the bottom! It has a double-wall as it passes through the tent. Some use rubber flashing instead.
Space around the stove. We leave a knife dangling at the opposite side to the door so someone can cut their way out.
Floor material. Our is a tough poly material, but it has zips to pull it back away from the stove (revealing the ground). It gets pretty warm if straight on the floor and the legs are approx 8″.
I wouldn’t be as worried about fumes as some here, especially if it’s a good airtight system with a tall flue to draw the air through.
Stoner on here, got a good cheap portable stove for his shed – look for pics. Not lightweight though.
Here’s mine (with only half the flue in place):
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Fourdog titanium stove. First burn[/url] by thingswelike[/url], on Flickr
Trying to find a pic of it in the tent