In the 1947 Republic serial, "The Black Widow," the title villainess on several occasions utilizes a rubber/latex-type face mask along with a wig to impersonate another character, disguising her voice and all, with the other actress taking her place in the proceedings.  Very identical to the scenario always presented in "Mission: Impossible."  Being a b-grade serial, I doubt it actually had much 'influence' at the time, but it clearly shows that the territory had already been explored.