The Lone Ranger theme sounds just like the third part of the William Tell Overture.
Ditto the theme song for Lark cigarettes. I remember one commercial (couldn’t find it on youtube, alas), where the Lark man was holding the “Show Us Your Lark Cigarettes” sign with the “Have a Lark, have a Lark, have a Lark today” chorus in the background.The Lone Ranger approachs the guy and asks what he thinks he is doing, using his music. In turn, William Tell (complee with bow, arrow and apple) appears and says to Tell “I was going to ask you the same question.”
Very clever commercial.
The Godfathers’ Birth School Work Death and John Prine’s We Are The Lonely are the same.
76 Trombones and Goodnight My Someone are the same except for Goodnight My Someone is
3/4 slower.
I’ve been keeping a list ever since sci–fi fans tried finding a melody for “The Green Hills of Earth”:
SONGS THAT CAN BE SUNG TO THE TUNE OF “THE BALLAD OF GILLIGAN’S ISLAND”
Mickey Mouse Club Theme Song
Stairway to Heaven
GILLIGANS ISLAND - STAIRWAY TO HEAVEN - YouTube
House of the Rising Sun
Yankee Doodle
Pop Goes the Weasel
Peaceful Easy Feeling
I’d Like to Teach the World to Sing
Yellow Rose of Texas
Ghost Riders of the Sky
Rocky Top
Lion Sleeps Tonight
Tangled Up in Blue
Whiter Shade of Pale
Light My Fire
O Little Town of Bethlehem
Jingle Bells
God Rest Ye Merry, Gentlemen
Greensleeves
Jolly Old St. Nicholas
The First Noel
O Tannenbaum
It Came Upon a Midnight Clear
I’ll Be Home for Christmas
Take Me Out to the Ball Game
In-a-Gadda-da-Vida
Marines’ Hymn
Wabash Cannonball
America the Beautiful
The Internationale
Onward Christian Soldiers
Ode to Joy
Mack the Knife
A Hundred Bottles of Beer
Clementine
La Cucaracha
Semper Paratus
The Wearing of the Green
(The Rising of the Moon)
The Itsy-Bitsy Spider
I’ve Been Working on the Railroad
Sympathy for the Devil
Rollin’ Down to Old Maui
Acres of Clams
Bread and Roses
Sink the Bismarck
Forest Green (UK “Little Town of Bethlehem”)
Tomorrow Belongs to Me
Poems that apply:
Jabberwocky
Little Miss Muffet
Jack and Jill
Doctor Foster went to Gloucester
Little Jack Horner
Itsy Bitsy Spider
Mary Mary Quite Contrary
Anything by Emily Dickinson
In classic country/mountain music, The Great Speckled Bird http://www.youtube.com/watch?v=9jzlOI-8J_4, Wild Side of Life HANK THOMPSON - The Wild Side of Life - YouTube, It Wasn’t God Who Made Honky Tonk Angels Kitty Wells - It Wasn't God Who Made Honky Tonk Angels - YouTube and I’m Thinking Tonight of my Blue Eyes http://www.youtube.com/watch?v=q4tBmpRdZ94.
I guess they’ve always known about reuse/recycle in the hills. 
As well as “Heil Dir im Siegerkranz”, the unofficial national anthem of the German empire from 1871 to 1918.
After further research, I found out that the tune to “God Save the Queen” was virtually the proto-anthem for almost half of Europe, also including Norway, Russia and Switzerland.
Louis Jordan was a pioneer in committed recycling. At least three of his biggest hits–“All for the Love of Lil”, “Slender, Tender and Tall” and “I Like 'Em Fat Like That”–shared the same melody. Probably a lot more, too.
The Red Flag is also sung to the tune of the White Cockade.
Óró, Sé do Bheatha 'Bhaile lifted its melody from What Shall We Do With The Drunken Sailor?
As Astorian points out Irish folk songs often share melodies. Another example is The Moorlough Shore is often sung to the same air as Foggy Dew(the 1916 ballad). Patrick Kavanagh used the air of The Dawning Of The Day/Fáinne Geal An Lae for On Raglan Road too.
In the same vein, after scoring a big hit with Shake, Rattle and Roll, Big Joe Turner came up with Flip, Flop and Fly, basically the same song with different lyrics.
To make matters worse, Ray Parker Jr.'s *Ghostbusters *sound exactly like I Want a New Duck. 
At some gigs, I will play a James Blunt’s “You’re Beautiful” and Jason Mraz’s “I’m Yours” interchangeably… no one even notices it’s two different songs.
I thought of another couple.
Bob Dylan’s *With God On Our Side *has the same air as Dominic Behan’s The Patriot Game.
Bob Dylan’s When The Deal Goes Down has the same air as Where The Blue Of The Night Meets The Gold Of The Day a tune most famously sung by Bing Crosby.
And they ALL sound like “Pop Muzik” by M.
Many anthems are based on already extant airs. As others have said God Save The Queen’s air is used in several and the American anthem is based on an old British drinking song.
Finland and Estonia have the same national anthem, which gladdened the hearts of the Estonians whenever Finland beat the Soviet union in something.
Morning Has Broken, most famously done by Cat Stevens, uses tune of a traditional Christmas song Child In A Manger
Speaking of Irish melodies used in Church…
There’s an old Irish song about a pretty girl called “Star of the County Down” (some of you may have heard Van Morrison singing it with the Chieftains). In December, many church choirs sing a song aout Mary and St. Elizabeth, entitled “Canticle of the Turning,” which uses that same tune.
Huh, and all this time I had thought that “Child in a Manger” was to the tune of “Morning has Broken”, not the other way around.
And I hadn’t a clue that “Canticle of the Turning” was about Mary and Elizabeth: All of the versions I’ve heard have seemed more eschatological.