The Russian phrase spoken by Dan Aykroyd to the Russian agents is: "Chem men'she znaesh', tem lushche." This translates as: "The less you know, the better."
The movie was inspired by Bob Hope and Bing Crosby's "Road" movies, thus explaining Bob Hope's cameo appearance. Hope's cameo of less than 30 seconds playing a round of golf through a tent was his final appearance in a film.
In an interview on The Today Show (a.k.a. Today) (included on the Saturday Night Live: The Best of John Belushi), Dan Aykroyd stated that the movie that would eventually become Spies Like Us was intended to star Aykroyd and John Belushi. Belushi's death shortly after the interview obviously prevented his participation.
In 2009 "Family Guy" presented something of a sequel to the movie called Family Guy: Spies Reminiscent of Us, featuring Chevy Chase and Dan Aykroyd as themselves on another spy mission. They explain that President Reagan appointed them real-life spies after seeing the film.
The final scene with the chess pieces represents Eastern Europe being knocked down - this pre-dated the Revolutions of 1989 where Eastern European nations overthrew their Communist regimes, leading to the dissolution of the Soviet Union in 1991.
John Landis:
[Looking directly into the camera]
Aykroyd's character looks directly into the camera when he realizes how the launched missile can be successfully stopped.