The two NSA agents played by Hank Azaria and K. Todd Freeman in Grosse Pointe Blank are sort of a marriage of this trope and the traditional Salt and Pepper buddy cop pairing. Martin Q. Blank (John Cusack) and Grocer (Dan Aykroyd) also fit somewhat, although they are rivals.
Going off of the Men In Black reference above, those characters in general have a similar dynamic as this type of character, and the Agents are a particularly good fit. There's three of them instead of two, but that hardly matters.
The Twins from the The Matrix Reloaded probably also count. While they don't really exposit much, they do provide some amusing dialogue, talking to each other in an unconcerned deadpan while being shot, and taking part in a highspeed car chase.
Sydney Greenstreet and Peter Lorre were sometimes teamed up this way after The Maltese Falcon. Subverted by their cameo in Hollywood Canteen, where after Lorre asks a man if he "would like to step outside for a moment", he innocently tells Greenstreet "I was only going to ask him if he wanted a cigarette".
MasterBlaster, the duo who run Underworld in Mad Max Beyond Thunderdome ("They are a unit; they even share the same name"). Master is a mental giant with a body like a small child's; for Blaster, the reverse is true.
4-Lom and Zuckuss in Star Wars. The pair even loses face when forced to work with a third bounty hunter in the hunt for the Yavin Vassilika.
Possible Real Life example: Burke and Hare. Their cinematic versions in 1960's The Flesh and the Fiends definitely fit (and Donald Pleasance is particularly Faux Affably Evil as Hare).
Mr. Wint and Mr. Kidd, in the James Bond movie Diamonds Are Forever. They have a strange habit of dispatching everyone else in the diamond smuggling ring and tried to off James Bond three times: once by leaving him to meet a fiery end in an incinerator; again by leaving him to rot in a pipeline; and the third and final time by appearing to him in person disguised as the cruise ship's kitchen crew.
Leonard and Willie (Damon Wayans and Kadeem Hardison) in I'm Gonna Git You Sucka are two bad guys who work for Mr. Big and spend the movie trying to kidnap and murder, but mainly be idiots. One is dumber than the other...but only slightly. They always end up getting beat up and having to decide between going out the window or down the stairs.