I'm really interested in the opening cartoons in the opening sequence, this URL has a mostly complete list of the cartoons used, with screenshots:
http://theinfosphere.org/List_of_opening_cartoonsCopied and pasted it for convenience. This list will be updated when a unknown cartoon is found:
Season 1:
"Space Pilot 3000" Little Buck Cheeser (1937)
"The Series Has Landed" Baby Bottleneck (1945)
"I, Roommate" Baby Bottleneck (1945)
"Love's Labours Lost in Space" The Wacky Wabbit (1942)
"Fear of a Bot Planet" A Corny Concerto (1943)
"A Fishful of Dollars" Betty Boop's Crazy Inventions (1933)
"My Three Suns" The Impractical Joker (1937)
"A Big Piece of Garbage" A Corny Concerto (1943)
"Hell Is Other Robots" Betty Boop and Grampy (1935)
"A Flight to Remember" Clutch Cargo - "Space Station" (1959)
"Mars University" Pigs in a Polka (1943)
"When Aliens Attack" Daffy - The Commando (1943)
"Fry and the Slurm Factory" The Simpsons shorts - "Making Faces" (1987)
Season 2:
"I Second that Emotion" Fresh Hare (1942)
"Brannigan, Begin Again" Pigs in a Polka (1943)
"A Head in the Polls" Ko-Ko Needles the Boss (1927)
"Xmas Story" Fresh Hare (1942)
"Why Must I Be a Crustacean in Love?" My Old Kentucky Home (1926)
"The Lesser of Two Evils" Felix the Cat Trifles with Time (1925)
"Put Your Head on My Shoulder" Chick and Double Chick (1946)
"Raging Bender" Hollywood Capers (1935)
"A Bicyclops Built for Two" Hollywood Capers (1935)
"A Clone of My Own" Ko-Ko Needles the Boss (1927)
"How Hermes Requisitioned His Groove Back" Felix the Cat Trifles with Time (1925)
"The Deep South" Scrap Happy Daffy (1943)
"Bender Gets Made" Up to Mars (1930)
"Mother's Day" Bold King Cole (1936)
"The Problem with Popplers" Up to Mars (1930)
"Anthology of Interest I" Bosko Shipwrecked (1931)
"War Is the H-Word" Neptune Nonsense (1936)
"The Honking" Bold King Cole (1936)
"The Cryonic Woman" Up to Mars (1930)
Season 3:
"Amazon Women in the Mood" Unknown
"Parasites Lost" Aladdin and the Wonderful Lamp (1934)
"A Tale of Two Santas" Box Car Blues (1930)
"The Luck of the Fryrish" Boom Boom (1936)
"The Birdbot of Ice-Catraz" Bubbles (1922)
"Bendless Love" Little Ol' Bosko in Bagdad (1938)
"The Day the Earth Stood Stupid" Unknown
"That's Lobstertainment!" Box Car Blues (1930)
"The Cyber House Rules" Congo Jazz (1930)
"Where the Buggalo Roam" The Emerald Isle (1949)
"Insane in the Mainframe" The Mild West (1947)
"The Route of All Evil" Heep Hep Injuns (1949)
"Bendin' in the Wind" A Coy Decoy (1941)
"Time Keeps on Slippin'" It's the Cats (1926)
"I Dated a Robot" Censored (1944)
"A Leela of Her Own" The Goal Rush (1932)
"A Pharaoh to Remember" Toys Will Be Toys (1949)
"Anthology of Interest II" A Close Call (1929)
"Roswell that Ends Well" Congo Jazz (1930)
"Godfellas" Unknown
"Future Stock" Jingle Jangle Jungle (1950)
"The 30% Iron Chef" Toys Will Be Toys (1949)
Season 4:
"Kif Gets Knocked Up a Notch" It's a Greek Life (1936)
"Leela's Homeworld" In a Cartoon Studio (1931)
"Love and Rocket" In a Cartoon Studio (1931)
"Less than Hero" The Dover Boys (1942)
"A Taste of Freedom" The Queen Was in the Parlor (1932)
"Bender Should Not Be Allowed on Television" Much Ado About Mutton (1947)
"Jurassic Bark" Hiss and Make Up (1943)
"Crimes of the Hot" Much Ado About Mutton (1947)
"Teenage Mutant Leela's Hurdles" Moonlight for Two (1932)
"The Why of Fry" Much Ado About Mutton (1947)
"Where No Fan Has Gone Before" Hiss and Make Up (1943)
"The Sting" The Queen Was in the Parlor (1932)
"Bend Her" Naughty But Mice (1947)
"Obsoletely Fabulous" Much Ado About Mutton (1947)
"The Farnsworth Parabox" The Queen Was in the Parlor (1932)
"Three Hundred Big Boys" Flop Goes the Weasel (1943)
"Spanish Fry" Much Ado About Mutton (1947)
"The Devil's Hands Are Idle Playthings" The Futurama opening itself.
Season 5:
"Bender's Big Score Part 1" "Space Pilot 3000" (1999)
"Bender's Big Score Part 2" "Space Pilot 3000" (1999)
"Bender's Big Score Part 3" "Space Pilot 3000" (1999)
"Bender's Big Score Part 4" "Space Pilot 3000" (1999)
"The Beast with a Billion Backs Part 1" The Soup Song (1931)
"The Beast with a Billion Backs Part 2" The Futurama opening itself.
"The Beast with a Billion Backs Part 3" The Futurama opening itself.
"The Beast with a Billion Backs Part 4" The Futurama opening itself.
"Bender's Game Part 1" Quasi at the Quackadero (1975)
"Bender's Game Part 2" None
"Bender's Game Part 3" None
"Bender's Game Part 4" None
"Into the Wild Green Yonder Part 1" And the Green Grass Grew All Around (1931)
"Into the Wild Green Yonder Part 2" None
"Into the Wild Green Yonder Part 3" None
"Into the Wild Green Yonder Part 4" None
Season 6:
"Rebirth" Unknown
"In-A-Gadda-Da-Leela" None
"Attack of the Killer App" None
"Proposition Infinity" None
"The Duh-Vinci Code" None
"Lethal Inspection" None
"The Late Philip J. Fry" None
"That Darn Katz!" Scotty Finds a Home (1935)
"A Clockwork Origin" None
"The Prisoner of Benda" None
"Lrrreconcilable Ndndifferences" None
"The Mutants Are Revolting" None
Counting that, 4 episodes (when I first posted this, 14) are currently missing any info on the original cartoons. To see snapshots of the opening cartoons of these episodes, please see the URL I posted. They are:
1. "Amazon Women in the Mood"
2. "The Day the Earth Stood Stupid"
3. "Godfellas"
4. "Rebirth"