For lack of a better name, let's call them supercuts. (Thanks, Ryan.)
Here are some examples I could find, but I'm sure there must be more. Post 'em in the comments and I'll add them. Bonus points for supercuts with the most clips, the shortest clips, and in additional genres (sports? politics?).
UPDATE: This entire list (and many more) is now on Supercut.org. Add more there!
Glengarry Glen Ross - Obscenity Count
Glengarry Glen Ross - "The Leads"
Ship, Computer, and Sauce in the first six Star Trek films
Shia LaBeouf in "No No No No"
Requiem for a Dream, montage of every drug montage (meta!)
Casino, every "fuck"
Big Lebowski, every "fuck"
Big Lebowski, Every "dude"
Big Lebowski, Every "man" (thx, Matt)
True Romance, every kill from the finale
True Romance, every "fuck"
Rushmore, handjob references (thx, Matt)
Scarface, every "fuck" (thx, oscar)
The Departed, every "fuck" (thx, oscar)
Jay and Silent Bob Strike Back, every "fuck" (thx, MeatFarley)
All the pauses and silence in His Girl Friday from 1940 (thx, progosk)
The Incredibles, buttons, doors, and explosions (thx, Joshua)
Fargo, every "yeah" (thx, Doobybrain)
Charles Bronson Death Wish Bodycount (thx, Dave)
Chris Hefner's Talking Picture (The Road to Ruin), removes all words from a 1938 film (thx, Jamie)
Boondock Saints, every "fuck" (thx, Brandon)
Midnight Run, every "fuck" (thx, Lakawak)
Miscellaneous films, cops turning in their badge and gun (thx, Darrin)
Every "McFly" from the Back to the Future trilogy
Cell phone cliches in horror films
Nearly every Hitchcock film cameo (thx, Oscar)
"We've Got Company" (thx, Buzzfeed)
"Get out of there!" (via Buzzfeed)
"It's Gonna Blow!"
Every lightsaber ignition/retraction from the Star Wars series (thx, Buzzfeed)
Everything Chris Klein Says in "The Legend of Chun Li" (thx, Jeff)
"You look like shit."
Every Arnold Schwarzenegger scream (thx, Buzzfeed)
Every Nicolas Cage freak-out
In 3D! (thx, Joe)
Every "Michael" in The Lost Boys
"Now if you'll excuse me..."
"Blah blah blah."
7 minutes of face slaps in film
"Nooooooooo!" (thx, Scott)
Every Bone Steven Segal Has Broken
Every "What?" from the LOST series
Legend of Zelda TV show, Excuse Me, Princess!
Knight Rider - Turbo Boosts
CSI: Miami, Caruso's One-Liners
The Simpsons, every couch intro
The Sopranos, Every single whacking
Ojamajo Doremi, every transformation
Star Trek: The Next Generation, Last 10 seconds of every episode of season 1
Every Dragonball Z transformation
Mighty Morphin' Power Rangers, every finishing move from Season 1
Deadwood, every curse in Episode 1, 2, and 3 (thx, Brian W)
LOST, Sawyer says "Son of a Bitch" (thx, Jordan)
Rozen Maiden, every "Desu" (527 times! More context, thx Jason)
Buffy the Vampire Slayer, every mention of "Buffy" from Season 1 (Chuck Jones)
Battlestar Galactica, every "frak" from season 1 (thx, Nowak)
Big Brother's Julie Chen, every "but first" (thx, Cardhouse)
House M.D., every "lupus" reference (thx, engtech)
LOST, Sawyer's nicknames in the first three seasons (thx, oscar)
Rachael Ray, "Mmm!"
MTV's Newport Harbor, 82 "like"s in one episode (thx, Jamie)
Red Dwarf, every "smeg" reference in all 52 episodes (thx, arto)
LOST, Desmond saying "Brother" (thx, cypher)
The Wire, Clay Davis' "Sheeeeeit" (thx, dunk3d)
24, Jack Bauer says "damn it" (thx, jonathan)
The Simpsons - Homer's D'ohs, excerpt from an official episode (thx, Scott)
Scrubs - Every Girls Name to J.D. from Dr. Cox (thx, oscar)
Naruto - Every Rasengan from the filler story arcs (thx, Binkley)
The Hills, without the dialogue (thx, Rex)
Bea Arthur says "Whoa!" on the Golden Girls (thx, BWE)
The Office - "That's What She Said" (thx, Jason)
"I'm not here to make friends" from every reality TV show (thx, Rich)
Dr. Who - "What are you doing here?" (thx, Duncan)
The Sopranos, every profanity
The Royle Family, every "My Arse!" (thx, Dunk)
Stargate SG-1's Teal'c says "Indeed" (thx, Torley)
Kramer's Entrances from Seinfeld (thx, Buzzfeed)
Every "Dude" from Hurley on LOST
Every Secret Ingredient from Iron Chef America (thx, Oscar)
Mad Men's Don Draper says, "What?"
Every cigarette smoked on Mad Men
Every "Christmas" from the 90210 Christmas special
Star Trek, Spock says "Fascinating" (thx, Scott)
MST3K, The Many Names of David Ryder (thx, Nathaniel)
Various films, Wilhelm Screams
Various films, The Slow Clap
Various films, NOOO!
Steve Jobs says "Boom!" in Mac keynotes (thx, Jeff)
Clip from Christian Marclay's "Telephones" from 1995 (thx, progosk)
iPhone "Hello" ad, heavily inspired by "Telephones" (thx, progosk)
Phil Collins' "In the Air Tonight," every "Oh Lord" (thx, oscar)
2008 State of the Union, without any speech
Bill Gates says "Uh..." in Rocketboom interview
Collage of "Zoom In and Enhance" scenes
"Hey, Guys!", collection of YouTube beauty guru intros
Collection of "Myspace salutes" (more context)
"I Could Tell You, But Then I'd Have to Kill You"
"We're not in Kansas anymore."
Club Drunk Party Hands, every use of those four words in the Billboard Top 10 for 2010
All the pauses in Sarah Palin's response to the 2011 Arizona shootings
Obama says "spending"
Only the inhaling from an hour of NPR's All Things Considered (Chuck Jones)
Loveline, every question asked by the hosts (Chuck Jones)
Loveline, every name mentioned in alphabetical order (Chuck Jones)
Loveline, every "Yes" and "No" (Chuck Jones)
Loveline, every number (Chuck Jones)
Loveline, interjections (Chuck Jones)
Loveline, dead air (Chuck Jones)
NWA's Straight Outta Compton, obscenities only (thx, oscar)
The "Most Obsessive" award goes to artists Jennifer & Kevin McCoy for their work Every Shot, Every Episode from 2001, a 277 DVD set compiling 10,000 clips from Starsky & Hutch, arranged by categories like "Every Dead Body," "Every Mirror," "Every Gunshot," and "Every Affirmative Response." Other work of theirs includes I Number the Stars, a shot-by-shot index of the first 20 Star Trek episodes in 120 categories, How I Learned, a 10,000 shot inventory of the show Kung Fu in over 100 categories, and Every Anvil, cataloguing the violence in 100 Looney Tunes cartoons. Mind-blowing. (Thx, Buzzfeed!)
June 15, 2011:
The earliest supercut appears to be Joseph Cornell's "Rose Hobart", a short film from 1936. It compiles nearly every scene featuring the obscure actress in the 1931 B-movie, East of Borneo.
Want more? Michael Bell-Smith and I launched a site dedicated to the supercut genre at Supercut.org.