I've heard this rumour before - I think it was started in a biography that came out last year. Apparently the Duke's movies were so anti-Communist and yet so popular, they ordered a hit on him to try and stop him from giving Communism a bad rap. The FBI caught the assassins before they could get to the Duke, and supposedly took the Duke and the Ruskies out to a secluded location where the Duke acted like he was going to kill the two of them. Alas, it was all for show, he just wanted to scare them.
The Jeffrey Wells column linked above is where I first heard about this. (It's the second story on the page.) I guess Orson Welles told the story at a party in 1983! There's a lot more information in the column, and an amusing photoshop of Tom Hanks as John Wayne since that's who [Jeffrey] Wells thinks should play him in a movie version of the episode.
Sort of related to this thread topic, Milos Forman tells a funny Stalin/movie story on the Amadeus Director's Cut DVD commentary. He says it was the inspiration for the scene in the movie where the Emperor yawns at the premiere of The Marriage of Figaro, which dooms it to a short run. Salieri explains that, "three yawns, and it would have closed that night. Two yawns, within a week. One yawn..." [cut to Mozart complaining to Salieri] "Nine performances! That's all it's had!"
The story about Stalin was that his underlings would watch him at screenings of new movies, when they were deciding whether or not to release them, censor them, etc. One yawn from Stalin, and they'd release the movie, but not give it much publicity. Two yawns, the movie would be banned. Three yawns, the director was sent to Siberia.