You really don't need a canvas tent for using a stove. Check out Kirafu, they make sil nylon tents that are stove compatible. You might want to do something DIY. A sturdy winter tent with a stove jack would be an awesome winter shelter.
You mentioned just bringing a "winter tarp"? Do you mean a flat tarp? I can tell you that sleeping in an open tarp with a stove will not work. You need a big, open fire for that. A stove will only keep you nice and warm if it's inside an enclosed shelter.
In normal circumstances, there isn't a fire hazard with nylon or sil nylon. It's not going to burst into flames unless it has direct contact with flames. What happens is the firewood pops and sends sparks into the air. If those sparks fly into thin nylon, it can create tiny little holes. This isn't an issue with a stove because it's closed off but is an issue with an open fire.
If you are camping with an open fire to heat your tarp shelter, you should have a cheap or cheapish tarp. It will get a few holes over time but you can just patch it up. You will get decent life out of it considering the cost. You are also going to want a cheap bivy (i am going to experiment with tyvek) to block any sparks from getting to your expensive down sleeping bag.
The amount of sparks flying around depends on the type of wood. I have slept extremely close to a fire with oak wood and there were no pops or sparks. On the other hand, I have slept next to a lot of large fires using coastal redwood and that stuff pops like crazy. I had a coal, about double the size of a quarter, fly through the air from about 5 feet away and land right in between my feet. It burnt straight through my poncho liner and all the way to the ground through my thermarest z-lite.
While you have to stoke a stove constantly, if you make a big, long log fire and stoke it right, you might not have to feed it again for several hours. If you do it enough, you pretty much start sleepwalking over to the wood pile and fall asleep instantly when you get back in your bag.