Would a chunky mantel allow me to mount my TV above a wood burner?
A few issues with your drawing:


code may very well not allow a wooden mantel directly above a woodburning stove like that. Woodburning stoves get extremely hot. Check the spec's on your stove to determine the minimum clearance from combustibles. 
Soot. No matter how hard you try, a woodburning stove is likely to kick a bit of soot into the room now and again. Not the best for TV screens. 
your neck may not like the angle. Lots of people put TVs above fireplaces and they look really great in magazines, but are often the worst possible place to actually place a TV. 


(BTW, very nice sketch! That's always a great asset to questions here.)