A bit off-topic, but I sing some words to tunes from some songs. For example:
[ul][li]Whenever I cross the Nooksack River (twice a day) I think of The B-52’s Love Shack[/li][li]Chuckanut Road reminds my of Holiday Road from National Lampoon’s Vacation[/li][li]The community of Semiahmoo reminds me of The Pogues Dirty Old Town[/li][li]“Phenomenon”, either the word or the film, gets sung to Mannah, Mannah[/ul][/li]And just for good measure, whenever I pass a statuary yard I put on a Quasimodo voice and shout “Statuary! Statuary!”
“Batman” can be sung to almost any blues song.
Listening to my Pure Moods II cd recently, I realized that “On top of spaghetti” can be sung to “Chariots of Fire.” Makes it sound all dramatic and stuff. Fun. Of course, now I can never listen to the latter without inserting the words of the former.
I regularly sing “Volare” to the tune of “Moon River” , and vice versa.
Not quite the same thing, but the band Orbital does a great combo version of Belinda Carlisle’s “Heaven is a Place on Earth” and Bon Jovi’s “You Give Love a Bad Name.” Uncanny.
FYI, Deutschland Uber Alles, more properly known as Das Lied der Deutschen or Das Deutschlandlied, written in 1841, hijacked the melody from a piece composed by Franz Joseph Haydn in 1797. The hymn I Am Waiting for the Dawning is attributed to S. Trevor Francis ~1835 continues in popularity, being found in some current hymnals.
There was (and may still be) a band called Big Daddy. They took current hits and performed them in the style of 50’s & 60’s tunes.
Examples:
“Welcome to the Jungle” = “The Lion Sleeps Tonight”
“Ice Ice Baby” = “Johnnie B. Goode”
“…Jungle” will bring (laughter) tears to your eyes the first time you hear it.
I like singing “Amazing Grace” to “(Ghost) Riders In The Sky.”
Sir Rhosis
I was unaware that “Batman” had any words other than “BATMAN!” You know, duh na na na na na na na, duh na na na na na na na, BATMAN! Elfman’s Batman theme is an instrumental. Unless there is a different song that goes by that name, color me 
I always enjoyed conflating “If I Had a Million Dollars,” by the Barenaked Ladies, with “If I Had a Rocket Launcher,” by Bruce Cockburn.
If I had a rocket launcher
If I had a rocket launcher
Some son of a bitch would die
But not really, that would be cruel
Amazing Grace, how sweet the sound, that saved a wretch like me –
A-M-A, Z-I-N’, G-R-A-C-E!
I once was lost, but now I’m found, was blind but now I see –
A-M-A, Z-I-N’, G-R-A-C-E!
‘Mazin’ Grace,
‘Mazin’ Grace,
Forever let us hold your head up high! High! High! High!
Yeah, I can see that …
The ABC Song
Twinkle Twinkle Little Star
Baa Baa Black Sheep
Beat me to it, dude. I sure hope JBJ sued somebody’s ass for many dollars over that one.
For me, it was the opposite. I’d never heard Chariots of Fire until recently, and when I did, I thought “This is famous movie music? It’s just a dramatic version of On Top of Spaghetti!”
I knew it was also a hymn, but not the Haydn connection: any link with the original to Bach, given that A Whiter Shade Of Pale pretty much plagiarised him?
I had never heard Chariots of Fire before (at least, not that I was aware of), so I checked around for a RealAudio sample. And may I just say
HA ha ha ha ha haaa ha ha!!! 
Amazing Grace can be sung to just about anything.
Similarly, lots of stuff can be sung to the Gilligan’s Island theme.
The 80s hair band Poison – came out with Every Rose Has Its Thorns, and then later they did Something to Believe In. These are essentially the same song (slight melody change, different words, same accompaniment works on both). Play one and sing the other.
While visiting Ireland early this August I heard a new single on the radio whose chorus maps perfectly onto the chorus of “Satellite of Love” by Lou Reed. Wish I knew who did the newer single, it was quite pretty despite the (probably unintended) coincidence. The vocals were female and kind of breathy and the song overall was quite lush and pretty.
“Built for Speed” and “On the Road”, both by Motorhead, are essentially identical. If there’s a reason why, I’m unaware of it. Not that I’d ever criticize Lemmy & co.! 
Sounds like there might be kind of a thematic similarity in there, too. 
The Tragically Hip song Scared and Amazing Grace.
(my good friend Eric gave me a cute demo of this a few years ago…)