Not really. The episode I just blogged, "The Return of Happy Spangler", contains a direct reference to John and Jackie Kennedy. I recall another, later episode in which Rob is clearly imitating Lyndon Johnson. Other contemporary references I can think of, just off the top of my head, include:
Rob going to the movie "The Guns of Navarone"; Millie alluding to the movie "Psycho"; Richie's naming his favorite ballplayer, Mickey Mantle; mentions of celebrities including Danny Thomas, Marilyn Monroe, Ingrid Bergman, Rock Hudson, Sophia Loren; Rob's brother, Stacy, naming rock stars Elvis Presley and Ricky Nelson; references to TV shows of the time including "The Twilight Zone", "I've Got a Secret", and "Candid Camera". No doubt there are many other examples.
Even without these direct references, the show is full of images and plot elements that identify the time period. The clothing and hair styles, the furniture in the Petrie home -- and the mentions of money. Buddy buys yogurt for a quarter; a couple can dine in a nice Manhattan restaurant for $12, including tax and gratuity.
So Reiner's claim is spurious; but who cares? One thing about the show is indeed timeless: Its comedy.