Are you worried about sparks? What type of tarp do you have?
Would you consider using a wood stove near your tarp?
I worry about sparks but have never had a problem. I cook under my tarp whenever it's raining, there's usually 4+ feet of clearance between just set up the tarp so you have clearance and take extra care. I make a point to not take the pot off with it going full bore. Just practice managing the stove well without the tarp in the picture first, and hey the fuel is free!
"don't ask yourself what the world needs. Ask what makes you come alive and go and do that because what the world needs is people who have come alive."
I've boiled water under my tarp, but never cooked. I don't want that stank on my tarp. The last thing I want is a bear stopping by to tell me my tarp is tasty. Definitely never had a fire under any tarp (except my cheap, leaks like a sieve anyway, Guide Gear 12 x 12 tarp).
"A foolish consistency is the hobgoblin of little minds." Ralph Waldo Emerson
I have never used wood stoves under my tarp but I do use the canister stove under it. I think using it near your tarp would be fine as long as you dont get crazy with a lot of wood! It would also probably be better to have your tarp in lean to mode to help vent.
Boing water with a small alcohol stove or a canister stove where you have consistent flame and or good control over it are one thing. Wood burners are another. Less control, chance of sparks and popping from pitchy wood all pretty clearly say no to use under or close to your gear. The question of open flame and or open fire under or near a hammock set up comes up a few times a year. You will get a few folks that do it and a bunch of us that say no. None of you hammock gear is flame resistant. It will all happily burn into a flaming lump of molten plastic right around you with very little encouragement. It does not take much of a flare up or group of sparks to turn your tarp, hammock, quilts pack and anything else handy at least into swiss cheese with little burn holes if not total meltdown. Add in the food smell issues mentioned above and most folks will give the idea a big
Just an out of shape middle aged guy who loves doing outdoor things with his great kids...
Boil water for coffee or tea in the morning with alcohol stove only.
We don't have bears or many larger apex preds, but I still like to find a nice place 100 yards or better from my campsite to cook and eat meals, then of course I throw that all away by hanging my food bag, Sargent Rock style, from the end of my hammock suspension.
That way I can fight the coons and mice off, from a comfortable position.