The reason the actual explosion of Phelps' car isn't seen on screen is because the car exploded prematurely. According to Patrick J. White in The Mission: Impossible Dossier, this was the first of very few backstage accidents on the series. Prop man Bill Bates said, "The trunk lid of the car, cabled down so it wouldn't fly too far away, snapped the cable and landed next to the camera, half a block away." "Boom!" recalled Peter Graves, "the fender went sailing past my ear and landed about fifty yards away. How nobody was hurt or killed, I don't know." White reports the effects assistant responsible for the wiring mistake was "reassigned, ultimately fired, and was killed rigging another exploding car for a commercial a few years later."
Before becoming an actor, Robert Phillips, who played the ruthless mob killer Johnny, was an undercover policeman whose exploits were the basis of the 1959-60 series Tightrope (1959) starring Mike Connors. Phillips' chilling performance as Johnny in "The Council" earned him a place on the preliminary Emmy nomination ballot for Best Supporting Actor. (Source: The Complete Mission: Impossible Dossier by Patrick J. White)
The sound of the elevator starting during the last scene was the same as the photon torpedoes being fired from the Enterprise in Star Trek TOS.
Paramount's "New York Street," where Phelps' car was "tailpiped," is the same street where Kirk and Spock got a young boy to act as a distraction in A Piece of the Action (1968).