Red Grant has inspired a long list of Aryan-looking near-undefeatable henchmen (whom Bond always manages to defeat). He's been imitated with Hans, Necros, Mister Stamper, and other Aryan-looking villains. It's arguable that Rocky IV's Ivan Drago is also meant to evoke Red Grant.
If your story has a shady leader of a super villain organization, played straight or parodied, chances are they were based on Blofeld. Dr. Claw from Inspector Gadget and Giovanni in his early appearances in the Pokémon animé are among the more obvious imitators and an episode of The Powerpuff Girls has another one as a one-off villain (which later reveals that the cat was the actual villain). And let's not get started with Dr. Evil.
I Am Not Spock: For the rest of Lotte Lenya's life, new people tended to look at her shoes.
Name's the Same: Red Grant's first name in the film is Donald, and thus he is a namesake of another British traitor, although the real one was a Nazi rather than a Communist.
The Other Darrin: Major Boothroyd, aka "Q," goes from being played by Peter Burton to the much more familiar Desmond Llewelyn.
Serendipity Writes the Plot: Rosa Klebb was fighting James Bond using a poisoned shoe knife. The script called for her to be accidentally killed by her own weapon, but the director couldn't figure out a way to film it that didn't look ridiculous. Then someone realized that a) there was a gun on the floor from when Bond had disarmed Klebb and b) the heroine, who had been an enemy agent recruited by Klebb before falling in love with Bond, was just standing there watching the fight. So the director changed the script to have the heroine pick up the gun, and after some hesitation, shoot Klebb.
Troubled Production: Pedro Armendariz (playing Kerim Bey) was critically ill and had to shoot all his scenes within two weeks. The script was constantly being rewritten all the way through. Major special effects (such as the wall of fire in the boat scene) failed putting the production desperately behind schedule. Director Terence Young and art director Michael White were nearly in a helicopter accident that could've killed them both. Sean Connery himself was almost killed while filming the scene where Bond is chased by the SPECTER thugs in the helicopter when the inexperienced helicopter pilot flew in too close and almost took his head off. It's a tribute to the professionalism of the cast and some extremely clever filming by editor Peter Hunt (a lot of Rosa Klebb's dialogue was rewritten and refilmed simply by having Lotte Lenya perform against a back projection) that the film survived and became so successful.
A tragic variant: like it says on the main page, From Russia with Love was chosen as the next film after John F. Kennedy said it was his favorite Bond book. The advance print arrived at the White House on November 21, 1963. Advisers thought that Kennedy would enjoy the film, after he returned from his trip to Texas....
Most sources say he did get to see it; he'd had a private screening the night before departing for Texas.