Firstly, what does the expression even mean? Heigh-Ho, according to the dictionary, is an exclamation used to express boredom, resignation, or jollity. You've got to figure in a film like Snow White, it is going for the latter.
In the film, the dwarfs sing the song as they finish working in the mine for the day and begin the journey home (wouldn't it be lovely to have a theme song to end the day with? rather than the honking horns of traffic?). The song was written by the team of Frank Churchill and Larry Morey.
As I found with Whistle While You Work, there are many references to this song in unexpected places:
1940: Donald Duck sings this song in "The Riveter".
1963: The song appears, with slightly different lyrics, at the finale of Walt Disney's Enchanted Tiki Room. I'm thinking that it hasn't always been the exit song. If you know when Disney started using it, let me know!
1984: In the movie, Gremlins, the Gremlins are watching the "Heigh-Ho" scene in a theater in Kingston Falls, and singing along.
1988: In Oliver and Compay, Tito sings "Heigh-Ho, Heigh-Ho, it's off to work we go" when he is rescuing Jenny.
1990: In the book, "Fudge-a-Mania", the character Sheila was singing the song after getting a baby-sitting job.
2004: In World of Warcraft,
when playing as a male Dwarf, characters can tell a joke in which the
character will sing the song. "Heigh-ho, heigh-ho... ugh, second verse,
same as the first."
2009: In a Simpsons episode, called Four Great Women and a Manicure, characters sing another parody of the Heigh-Ho song, "Ho-Hi, Ho-Hi".
2010: In the last scene of Scared Shrekless the song can be heard as the
dwarves silhouettes are seen on a rock as the dwarves walk along shortly
before they are egged by Shrek and the gang
2010: A version of the song recored by Bunny Berigan and his Orchestra were used in a Levi's commercial
2012: In the film Snow White and the Huntsman, a dwarf named Beith "Hi-ho lads, it's off to work!" and another dwarf threatens to "smash his face in" if he starts whistling.
2013: Saving Mr. Banks features the song as P.L. Travers arrives to the Disney Studio and is taken to Walt's Office.
2014: In ABC's Agents of S.H.I.E.L.D, the song is sung by Leo while using the D.W.A.R.F. drones.