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Guessing it is from "toff", a thupping English term referring to the genteel, and maybe, from the practice of the gentry wearing wigs. Perhaps us colonized persons then doffed the "toff-" to portray the English in our plays. And then the term probably came to be used for wigs in general. (In the famous words of Wikipedia, citation needed.)
Tofuns are used largely by moving theater groups. They're ghastly looking and smelly, and the strong gum hurts when you pull them off. They're used especially by imitation Shivas, munis, Galileos alike. Very popular during Dusshera.