I'm assuming your firemarshall won't allow anything really burning.

Paint Franks real thumb black so it looks singed.

Place an air bladder under Frank's armpit, tube running through his jacket into fake thumb. Fake thumb has several small pieces of red & orange silk stuffed in it and hole in end--or flap in end. Frank squeezes bladder when 'ignited' forcing silk to blow out of thumb. Of course he freaks out and 'flame' goes higher (he squeezes more). And Frank dances around putting out his thumb.

Once Frank 'puts out' thumb: he really has stuffed fake thumb into his coat sleeve, revealing real thumb which is painted black.

Cheap air bladder can be hot water bottle or some soccer ball/air matress foot pumps work well.