I can only think of two - “Doh, a Deer, a female deer..” and the chorus of Leonard Cohen’s Hallelujah, “It goes like this / the fourth, the fifth / The minor Fall / The major lift..”. There have got to be more. Extra points for non-didactic intent, like the second example.
This is the song that never ends
Yes, it goes on and on my friends…
A root, a toot, a toodlie-a-da-toot.
He blows it eight to the bar, in boogie rhythm.
He can’t blow a note unless a bass and guitar
is playin’ with him.
And the company jumps when he plays reveille.
He’s the boogie woogie bugle boy of company B.
Good one.
Second verse, same as the first!
I’m Hen-ery the Eighth I am …
[Herman’s Hermits]
sol do la re ti mi, a little slower,
and please keep the fingers curved as nice
and high as you possibly can …
[The Music Man: The Piano Lesson]
Lets take it to the bridge
RORY
Don’t get faster dear.
Let’s leave Shatner out of this.
Andy Fairweather Low did a song called “Hot Poop” in which the lyrics were the chord changes called out as they played. Can’t find a you tube vid or a lyric sheet anywhere..alas.
Slan,
D. 
For a seasonal entry, how about Jules Styne’s “Christmas Waltz?”
It’s that time of year when the world falls in love
Ev’ry song you hear seems to say "Merry Christmas,
“May your New Year dreams come true”
And this song of mine in three-quarter time
Wishes you and yours the same thing, too
This morning I also remembered a song by 18th century American composer William Billings. There’s a RealAudio version here:
http://www.pittsburghcamerata.org/omi.htm
There’s a version by a high school choir here for those of you who, like me, think RealPlayer behaves more like a virus than an application:
http://www.youtube.com/watch?v=7pTtEOn6vMA
Modern Music (1781)
William Billings (1746-1800)
We are met for a Concert of modern Invention;
To tickle the Ear is our present Intention.
The Audience are seated
Expecting to be treated
With a piece of the Best,
With a piece of the Best.
And since we all agree
To set the Tune on E,
The author’s darling Key
He prefers to the Rest.
Let the Bass take the Lead
And firmly proceed,
Till the Parts are agreed
To fugue away.
Let the Tenor succeed
And follow the Lead,
Till the Parts are agreed
To fugue away.
Let the Counter inspire
The Rest of the Choir,
Inflam’d with Desire
To fugue away.
Let the Treble in the Rear
No longer forbear,
But expressly declare
For a Fugue away.
Then change to brisker Time
And up the Ladder climb,
And down again;
Then mount the second Time
And end the Strain.
Then change the Key to pensive Tones and slow
In treble Time; the Notes exceeding low
Keep down a While, then rise by slow Degrees;
The Process surely will not fail to please.
Thro’ Common and Treble we jointly have run;
We’ll give you their Essence compounded in one.
Altho’ we are strongly attach’d to the Rest,
Six-four is the Movement that pleases us best,
Six-four is the Movement that pleases us best.
And now we address you as Friends to the Cause;
Performers are modest and write their own Laws.
Altho’ we are sanguine and clap at the Bars,
’Tis the part of the Hearers to clap their Applause,
’Tis the part of the Hearers to clap their Applause.
“Generic Up-Tempo Folk Song” by The Limelighters
(can’t find lyrics, but you can hear it here)
Tom Lehrer’s “Folk Song Army”?
Final verse of “Robin Hood” by the Woods Tea Company".
And I quote.
Now Robin married Marian
They live out in the forest
happily ever after
Now here comes the chorus!
There’s the free floating verse that’s been attached to several songs
Second verse,
same as the first
A little bit louder
and a little bit worse.
~
It songs that sing their musical structure* that are harder to find, I suspect.
*By which I mean something you’d cover in a theory class. My example here wouldn’t be, but L. Cohen really is singing the changes.
Not exactly “singing their structure”, but still sort of musically self-referential:
A friend contributed this one:
How does it feel to be
One of the beautiful
People?
Tuned to a natural E,
Happy to be that way.
Now that you’ve found another key
What are you going to play?
Elton wrote the music
Lyrics: Bernie Taupin 
Somewhat in that vein:
Violins:
The violins ringing like lovely singing.
The violins ringing like lovely song.
Clarinets:
The clarinet, the clarinet
Goes doodle doodle doodle doodle dat.
The clarinet, the clarinet
Goes doodle doodle doodle dat.
Trumpets:
The trumpet is braying,
Ta ta ta ta ta ta ta ta ta ta ta ta ta ta ta ta ta.
Horns:
The horn, the horn
Awakes me at morn.
The horn, the horn
Awakes me at morn.
Drums:
The drums playing two tones
They’re always the same tones.
The drums playing two tones.
They’re always the same.
I learned the last verse as:
The big bass plays two notes,
And these are the two notes,
Five, one, one, five,
Five, five, five, five, one.
You sang the dominant pitch for five and the tonic for one.
May I play my cross-hand piece now, Miss emmline?