Congratulations to Laura who knew it was "Footsteps in the Dark" (1941) with Ralph Bellamy and Errol Flynn. Have a look for our previous discussion of the movie.
Dentists did not routinely wear gloves when I was growing up. I believe that practice started in the 1980s, possibly related to evolving hygienic protocols during the AIDS scare. Maybe some of you could dispute or corroborate that.
I don't suppose the intruments in Errol's mouth are sterile. (Not that they wouldn't have been in a real dentist's office, but I suspect the prop guy on the set did not posses an autoclave.)