There's friction somewhere, and it could be just a dirty, gunky chain.

Two things I can think of that would happen with a nasty chain -- it could actually make the RD "open up" (this could also happen with dirty jockey wheels) which almost erases tension on the chain and lets it fall off, or the chain could fail to unstick from the front chainring and get caught under the FD cage.

I can backpedal as quickly as I can front-pedal on my bikes and they're fine.

Not necessarily THE solution, but something to check for.