It sounds like Forever Fairytales (some of which I’ve seen) and yes, they were Russian-animated fairy tales dubbed into English. I think they came from the Soyuzmultfilm animation studio, which was the Soviet Union’s answer to Disney. They were dubbed into English sometime in the fifties or sixties and shown on TV under the Forever Fairytales name as part of certain kids’ shows. (In the original versions, many had songs, but the Forever Fairytales dubs took them out.) If it’s what I’m thinking of, the version of Beauty and the Beast was called The Scarlet Flower in its Russian version.
Soyuzmultfilm also brought us the 1957 animated version of The Snow Queen, which was dubbed into English by Sandra Dee, Tommy Kirk, Paul Frees, and others, with an intro by Art Linkletter. They ran that on TV a couple times when I was a kid.
The interesting thing is that these same fairy tales were dubbed into English AGAIN, with fairly big-name actors, for a series that Mikhail Baryshnikov produced called Stories From My Childhood. (This also included a new dub of The Snow Queen. That’s how I found out about Russian animation–I stumbled across this series on PBS one Christmas day, recognized The Snow Queen, looked up Soyuzmultfilm and found out about the earlier dubs.) It was pretty cool, but to me, the newness of the voiceovers and the new music soundtracks seemed to contrast too much with the old style of the animation.
You can find many of the Russian originals on YouTube. Is this the Beauty and the Beast you were thinking of?