Futurama episode Space Pilot 3000
March 28, 1999
At the very beginning of the series' premiere episode, a spaceship is seen flying through space, with Fry providing a voiceover. It is then revealed to the audience that Fry is actually playing an Asteroids-style arcade game, just as he encounters a gorilla similar to Donkey Kong, except with overalls. Although Fry attempts to blast the gorilla with lasers, the gorilla throws barrels at the spaceship, causing Fry to lose. A later shot of arcade cabinet then reveals that the game is entitled Monkey Fracas Jr., probably as a throwback to Donkey Kong Jr.
Futurama episode A Bicyclops Built for Two
March 19, 2000
The ninth episode of the second season of Futurama (Episode 22), opens with the Planet Express Crew going into the future's incarnation of the internet, and after a while they play a virtual reality game called Death Factory III. As Fry is seen running trough the game, he's attacked by a Donkey Kong-like enemy, that's then blasted by him. This enemy is actually almost identical to the one from Monkey Fracas Jr.
Futurama episode Anthology of Interest II
January 06, 2002
The third episode of the fourth season of Futurama (Episode 50) is about the Planet Express crew using for the second time the "What If" machine, showing three "imaginary" stories. The second segment, titled "Raiders of the Lost Arcade" has Fry asking the machine "What if life was more like a videogame?" and shows aliens from planet Nintendu 64 invading Earth. One of them is Donkey Kong, who's represented very similar to his original appearance, the most notable difference being that he wears a loincloth.
He's introduced near the beginning of the segment as "Ambassador Kong" by President Richard Nixon, who's signing a peace treat with the Nintendians in the United Nations building. Upon seeing him on TV, Fry says "Wait a second. I know that monkey. His name is "Donkey"!" to which Prof. Farnsworth replies "Monkeys aren't donkeys! Quit messing with my head!" Donkey refuses to sign the treat and instead throws a barrel at President Nixon, then escapes using ladders similar to the ones in the introduction for Donkey Kong. Once on top of the building he starts stoping, tilting the floors diagonally, similarly to the griders in the 25m stage. As all ambassadors are shocked, the Italian ambassador is then shown to be a character almost identical to Mario, who then runs away as the scene changes.
Donkey Kong returns in the ending scene, walking out of the Nintendian ship, and then asking the people of Earth to hand over a million of dollars in quarters. Lrrr then explains that the Nintendians need the quarters to do their laundry, and implies that Donkey Kong's loincloth needs washing.
Note that while characters directly interact with the ape, this can't be considered an actual In-Universe link because it's not authorized by Nintendo, meaning that it is not an actual incarnation of Donkey Kong, but just a character based on him, a parody. Since he's called by name, however, we can assume that Donkey Kong is in fact an actual game in Futurama's universe, making this a sub-universe link.
For other references in the episode see Raiders of the Lost Arcade references.
Futurama episode The Devil's Hands Are Idle Playthings
August 10, 2003
Theighteenth and final episode of the fourth season of Futurama (Episode 72), is about Fry swapping hands with the Robot Devil in order to be able to play the holophonor, an instrument that when played, creates holograms. While playing it during the episode, some dancers appear and climb a staircase, then some barrels start rolling down the steps, so they jump over them in the same style as Mario in Donkey Kong, even making the same sound, and making a score of 100 appear above their heads.
Futurama: The Beast with a Billion Backs
June 24, 2008
In a scene in the second Futurama movie, Fry visits an arcade and one of the cabinets is a parody of Donkey Kong called Honkey Kong. The title is even written in the same style, with stars in the "O"s, but it's colored differently.
This part of the movie was adapted into the fifth episode of the fifth season (episode 77) of the TV show.
Futurama episode Law and Oracle
July 07, 2011
At the beginning of the sixteenth episode of the sixth season of Futurama (Episode 104), Fry is seen playing a videogame clearly based on Paperboy. One of the obstacles is a barrel-throwing gorilla, clearly based on Donkey Kong. It comes out of an open manhole and wears overalls, like the first two Donkey Kong look-alike in previous episodes.
Futurama episode Reincarnation
September 08, 2011
The twenty-sixth and final episode of the sixth season of Futurama (Episode 114), features three different non-canon segments, each one of them animated in a different style. The second segment, titled "Future Challenge 3000" features a graphic style based on 8-bit graphics, and references to videogames. In a scene a depressed Professor Farnsworth can be seen wandering in a Super Mario Bros.-inspired setting and jumping over barrels, referencing Donkey Kong.
Fox, that produced Futurama, didn't own any right over Nintendo's Donkey Kong, but they only included minor references to the series.