This is a presumably incomplete list of movies whose soundtracks include different versions of the same song, one sung by characters in the movie, and a cover by a pop artist or artists, which usually (but not always) plays over the end credits. I will not be including songs that were recorded years later for "special edition" soundtracks. So far, all the movies on the list are animated, and almost all are from Disney. I'm not sure how many non-animated or non-Disney examples exist, but I assume some might exist. (Movies will be listed chronologically.)
1986: An American Tail (soundtrack): "Somewhere Out There", cover by James Ingram and Linda Ronstadt
1991: Beauty and the Beast: "Beauty and the Beast", cover by Celine Dion and Peabo Bryson
1992: Aladdin: "A Whole New World", cover by Peabo Bryson and Regina Belle
1994: The Lion King (soundtrack): "Circle of Life", "I Just Can't Wait to Be King", and "Can You Feel the Love Tonight", covers by Elton John
1995: Pocahontas: "Colors of the Wind", cover by Vanessa Williams
1996: The Hunchback of Notre Dame: "God Help the Outcasts", cover by Bette Midler
1997: Hercules: "Go the Distance", cover by Michael Bolton
1998: Mulan: "Reflection", cover by Christina Aguilera
2013: Frozen: "Let It Go", cover by Demi Lovato
2016: Moana: "How Far I'll Go", cover by Alessia Cara