Haven't seen the movie, and no, I don't believe he has a characteristic scar that I've noticed, and I do read the comic book. (Still, I know, one of my infectious hobbies.) Did find this on movie-mistakes.com though:
" Peter has a small horseshoe-shaped scar on his right cheek. In Dr. Octopus's lab, as Octopus is destroying the fusion reactor, they share a meaningful look and the scar has switched cheeks."