Looking for TV show or movie with monsters that came to life from old film reels

"Circle of Fear" (1972-1973 TV series)? Episode: Graveyard Shift

This sounds like episode 19, starring Patty Duke and John Astin, called "Graveyard Shift", although it's in color. The plot details seem to match up and the decade is the right one. It's also an anthology series as you remembered. This series only lasted 1 season. Below is a user review of this episode from IMDb:

Not the best the series had to offer, but not the worst either. It's a fun concept---the idea that classic film horror characters have come to life and haunt their old studio. Actually, I felt a bit sympathetic toward them! It's always enjoyable to see Patty Duke and John Astin (who were married in real life at this time), though the script doesn't really give them a whole lot to do. And I got a kick out of seeing series creator William Castle make a brief cameo as the head of the studio.

I remember finding this very scary indeed when I watched it as a eight-year-old, but re-watching as an adult I'm afraid the five film monsters just aren't all that frightening---they're more silly than scary, in fact. Overall, I'll give this an "eh..." sort of rating; it isn't terrible, but its primary charm is the result of nostalgia, rather than from being a solid, creepy story.

I didn't think they had too many episodes available, but I found this one on YouTube. You can watch the entire episode below, the film burning scene starts at about 46:50.