In the spy-genre there is ablsolutely nothing to compare to George Smiley and ‘Karla’ (the russian spy-master) from the Le Carre novels.
Peary and Cook? Amundsen and Scott? Newton and Leibniz? Cope and Marsh?
M and General Gogol, in the James Bond movies.
(After Bond destroys the gadget they both have been hunting for)
“That’s detente, comrade. You don’t have it, I don’t have it.”
(And Gogol just smiles, shrugs, and leaves.)
Spoilerized for anyone who hasn’t seen the movie:
Kid Shelleen and Tim Strawn (both played by Lee Marvin)
Two from related sitcoms:
Cheers and Gary’s Olde Towne Tavern were archrival bars/owners
Frasier Crane and his upstairs neighbor Cam Winston (Brian Stokes Mitchell) were archrival snobs and condo board members on Frasier (in the episodes when Frasier’s archrival wasn’t his brother Niles)
Nayland Smith and Dr. Fu-Manchu
Although Dr. Petrie manages by favorable circumstance to convince Dr. Fu-Manchu he is brilliant chemist and desirable acquisition.
I tried to do this a while ago in the form of a quiz. There’s a radio show I like that takes questions from listeners. I was going to call it “Name that Nemesis”; given the name of a character or person, name his more famous rival. Even started a thread, but I don’t think we ever got enough good ones to make a full category. My two favorites:
Charles DreyfusJacques ClouseauRalph WolfSam Sheepdog
Auburn and Alabama.
Godai and Mitaka from Maison Ikkoku.
Gau Ban and Scrib Lowengren from Shadow Skill.
Ayeka and Ryoko from Tenchi Muyo.
…thinking about it, I’m somewhat suspecting that the list is long in anime land…
Ron Burgundy and Wes Mantooth.
Koga and Inu Yasha
Macy’s and Gimbals
Spike Speigel and Andy 
Famous arch rivals?
Burger King
box girder
Reggie
Ha-ha-ha! Ha-ha-ha-ha-ha![/mandark]
Man, I loved that goofy toon when my son used to watch it.
Earl White Haven and Admiral Sonja Hemphill from the Honor Harrington novels. Proponents of competing military philosophies and rivals, they were both fundamentally after the same goal, victory in their war against Haven.
Godzilla and King Kong
Alien and Predator
Sylvester and Tweetie
David and Goliath
Professor Charles Xavier and Magneto.
West and Loveless
Smart and Siegfried
Seinfeld and Newman
Newton & Leibniz
A terrific example, which I had in mind when I wrote the thread, is **Scrooge **and Flintheart Glomgold in the old Barks’ comics. Seinfeld and Newman is a pretty good recent example, too.