What are some popular songs that are sung to the same tune?

Not exactly sung, but they do have a common melody: checkout Sinfonie Nr. 3 in F-Dur, Op. 90 – III. Poco allegretto by Johannes Brahms and then Love Of My Life by Santana (ft. Dave Matthews)

Dumb and Polly by Nirvana. Same tune; different lyrics.

“do your ears hang low” and the “do you hear the people sing” song form les miserable have really similar tunes – they sound good when sung at the same time

The chorus on Phish’s Farmhouse sounds an awful lot like Bob Marley’s No Woman No Cry.

ABC song and Twinkle Twinkle Little Star. Also, Knick Knack Paddy Wack and the Barney Theam.

Sorry I have kids.

Vanilla Ice borrowed the beginning of Ice Ice Baby from Queen’s Under Pressure.

Pretty much anything that’s freestyle, heavy metal, or especially pop/R&B.

Hmmm…Eric Hutchinson always does mash ups of songs that go to the same tune:
R Kelly’s Remix to Ignition/My Girl/My Humps http://www.youtube.com/watch?v=jfTxntx7r7A
Justin Timberlake http://www.youtube.com/watch?v=cZzB0m-rmKM
even different songs that go to the same tune of his own song!
http://www.youtube.com/watch?v=n3uunq8JTRk&feature=related

A few things use the traditional Shaker song “Simple Gifts.” Most notably: the Protestant hymn “Lord of the Dance,” the old TWA commercial, and “The Greatest Man Who Ever Lived” by Weezer.