For this the palm must be awarded to the Major-General’s song in The Pirates of Penzance. How about Tom Lehrer’s “Lobachevsky”?
I havd a friend in Minsk
Who has a friend in Pinsk
Whose friend in Omsk
Has friend in Tomsk
With friend in Akmolinsk. …
Any others?
“Not Getting Married Today” from Company, at the intended speed. There are quite a few other ones in Sondheim’s body of work, too.
How about “It’s the End of the World As We Know It” by R. E. M.? I’ve tried to sing along with that in a karaoke game and even with the lyrics right in front of you, it’s hard.
Also “Hardware Store” by Weird Al is in the same category.
And “One Week” by Barenaked Ladies.
“We Didn’t Start the Fire.” Apparently even Billy Joel himself dreads singing it live at concerts, because if you mess up one word, everything turns into chaos.
Danny Kaye practically made a career out of “patter” songs, like this one.
“Pick-a-Little, Talk-a-Little” from Seventy-Six Trombones.
“One Week” by Barenaked Ladies?
Psst–Post #3. 
I can’t bring YouTube sequences up on my computer (I only have Dial-Up); is that the ditty from Court Jester?
The Wake of Magellanby Savatage.
Particularly this bit during the chorus, which is sung very quickly:
Columbus and Magellan and De Gama
Sailed upon the ocean
In a world of ignorance
With thoughts so primitive
That men were killed
With no more will
Than that they simply had the notion
But in this world of heartless men
This thing they never did.
It appears to be from an album of selections from his radio show: Tschaikowsky and Other Russians.
Спасибо!
From “Old King Tut” a 1923 song. It’s sung at a fairly brisk pace, making the third line in the selected lyric a modestly impressive vocal feat.
“In old King Tut-Tut-Tut-enkhamen’s day,
the dancers then in style
would even make the old Sphinx smile…”
Along with “Jerry Springer” by Weird Al.
Nagasaki (I chose this clip because it has a clear rendition of the lyrics - not so easy to find.)
Gesundheit!
Hahahahahahahaha… 
How aboutLife is a Rock (but the radio rolled me)?
And there’s always the immortal I’ve Been Everywhere.
Who recorded that one? I’d like to find it… 