Dobutsu no Mori
April 14, 2001
The NES version of Donkey Kong and Donkey Kong Jr. Math are featured in the game as a playable game. It can be obtained in various random ways such as Tom Nook's Monthly Raffle, Crazy Redd's, as present from a neighbor, treasure hunts, birthday presents, etc. They return in all the revisions to the game (Dobutsu no Mori+, Animal Crossing and Dobutsu no Mori e+). See here for other NES games included in the game.
Additionally, the character Louie is a gorilla that looks very similar to Donkey Kong, and he's likely based on him. He also wears the Lil Bro Shirt, that has Luigi's logo on it. Louie also returns in some later games in the series.
Dobutsu no Mori+
December 14, 2001
While Donkey Kong and Donkey Kong Jr. Math return as base games obtainable at random locations, in Dobutsu no Mori+ and its revisions (Animal Crossing and Dobutsu no Mori e+), Donkey Kong Jr. and Donkey Kong 3 are added, but they're normally unaccessable and were made available only through Nintendo's giveaways.
See here for other NES games included in the game.
Animal Crossing-e Series 1
One card in the first series of Animal Crossing e-reader cards features a Mario design and it' called "Jumpman Mario", referencing Mario's original name from the Arcade Donkey Kong. Despite the name, Mario's sprite is actually taken from Super Mario Bros.
Donkey Kong Country: Tropical Freeze
February 13, 2014
During his idle animation, Donkey Kong is sometimes seen playing a Nintendo 3DS and sounds from various games can be heard, including Animal Crossing: New Leaf
Copyrights for both series are held by Nintendo.