Paul Simon’s “American Tune” has the same melody as the hymn “O Sacred Head, Now Wounded”
Mentioned by TruCelt in Post #9.
There’s a song called “The Riddle Song” (I gave my love a cherry without a stone/I gave my love a chicken without a bone) that shares a melody with The Twelfth of Never, as popularized by Johnny Mathis.
Ooh! Also O Tannenbaum and O Christmas Tree!
If you go that route, the Maryland state song (‘Maryland, My Maryland’) has the same melody as ‘O Tannenbaum.’ 
The German anthem “Deutschland, Uber Alles” was set to an old melody from a Haydn string quartet. The Columbia University alma mater uses
that same melody. With all the Jewish faculty and students at Columbia, I’m a bit surprised that wasn’t changed a long time ago (even though, obviously, Haydn wrote the tune LONG before the rise of the Nazis).
The “SpongeBob SquarePants” theme song uses the melody of the old sea chanty “Blow the Man Down.”
Lady Antebellum - “Need You Now” and Alan Parsons Project - “Eye in the Sky” sound like the same song to me.
Part of the melody of Elvis’ Can’t Help Falling in Love was lifted from the French song Plaisir d’Amour.
The melody of Country Joe’s Feelin’ Like I’m Fixin’ to Die Rag was copied from Kid Ory’s Muskrat Ramble.
The riff from In the Mood (made famous by Glenn Miller) is pretty much the same as that from an earlier tune by Wingy Manone called Tar Paper Stomp (sorry, I couldn’t find the original Wingy recording on line).
“Turkey In The Straw” and “Old Zip Coon” both use the same melody, also used to call herds of children to feed on ice cream.
Okay, the Vasectomy Song from Family Guy is sung to the tune of Coney Island Washboard.
Creedence Clearwater Revival’s Run Through The Jungle and John Fogerty’s* The Old Man Down The Road*.
At least that’s what Fantasy Records claimed when they sued Fogerty forplagiarizing himself…
And *Wooden Heart *comes from a German folk tune.
I’ve often thought that *Da Do Ron Ron *is a rip-off of Habenera from Carmen.
The Rhodesian national anthem - Rise, O Voices Of Rhodesia - is set to the melody of Ode to Joy. I never could understand why they didn’t try to compose a new melody. I don’t get it.
The guitars in the chorus of Boston’s More Than a Feeling are clearly playing Louie Louie.
Then on a slightly different note (!) **Music of the Night **from Phantom of the Opera –to me- sounds an awful lot like the classic novelty song “Fish Heads”.
Such that, “Fish heads, fish heads, rolly polly fish heads” can be sung note for note in place of each of the opening lines of the main stanzas.
Why Barnes and Barnes, the writers of “Fish Heads” didn’t sue the pants off old Andrew Lloyd, I’ll never know.
Only partially. The ‘Spongebob Squarepants’ response doesn’t scan to the ‘way hey, blow the man down’ or ‘give me some time to blow the man down’ lines.
But the pirate’s questions do scan to the first and third line of each verse.
the chorus of Bob Marley’s Buffalo Soldiers is the same melody of Banana Splits Tra-La-La
“Full Moon and Empty Arms” is the main theme of Rachmaninoff’s Piano Concerto #2, 3rd Movement.
“Tonight We Love” is the main theme of Tchaikovsky’s Piano Concerto #1, 1st Movement.
Procol Harum’s “A Whiter Shade of Pale” is very similar to Bach’s Air on a G String.
And of course all the songs from the show “Kismet” were based on music by Borodin.