simple plastic tarps will work fine for a basic tarp setup, especially if you are going bigger. they are a bit heavy and bulky, but not all that bad compared to tents with equal inside volume etc...also much more versatile.
i have considered, but not tried yet, using a small fire inside a tarp shelter. the plan i have in my head would be:
-dig hole about 12-16" deep, about 8" wide.
-start fire outside of shelter and get a good bed of coals going
-bring coals inside, place in bottom of hole. feed the coals with small twigs to keep them from dying.
with a tarp strung so that it is a closed up room (you would need a larger tarp) will allow you to capture alot of warmth inside...just make sure you have plenty of ventilation like was mentioned. the trick is to allow enough ventilation without letting in alot of cold air...
remember that fire inside any structure is very dangerous and should be carefully considered. also, i suggest never sleeping with a live fire in your shelter with you!