Happens a couple of times in Linus and Lucy (the Peanuts theme).
American Girl by Tom Petty has a cheesy disco interlude in the middle of it.
Similarly (but even odder), “Roll The Bones” by Rush is their typical song - except for the rap interlude dropped in the middle (at 3:21, to be precise)!
Thought of a Christmas classic, Brenda Lee’s “Rock Around The Christmas Tree”. That forceful saxophone.
Coincidentally, I ran across a classic example just yesterday, listening to the album Portraits by the Buckinghams, which I picked up last week. The song “Susan”, which made it to #11 on the Billboard charts, is a sunny pop song that has a totally incongruous psychedelic segment inserted at the 1:30 mark. It sounds kind of like the orchestral crescendos in “A Day in the Life”, and according to Wikipedia, it includes a short section of Charles Ives’ “Central Park in the Dark”. It was added, over the protests of the group, by their producer/manager, James William Guercio. Many radio stations edited that part out.
Classical:
Richard Wagner’s Tannhäuser Overture Middle section begins 5:32 and lasts until 11:55
The baseball play-by-play in “Paradise by the Dashboard Light” will throw most.
(it was done by a famous (to fans) announcer from a MLB team who provided it - he insists he had no idea it was to be used as a sexual metaphor - right, Meatloaf is doing a documentary on baseball, suuuuure…).
No, it’s Knights in White Satin. 
It’s about the finery the medieval King’s warriors wore while courting the ladies in between battles and dragonslayings.
The Chain by Fleetwood Mac
AKA worst song to slow dance to. How incredibly awkward during an 8th grade school dance, I get matched with the boy I have a crush on and we have to slow dance to The Chain. Sure fine, starts off slow, sway back and forth, not too bad and then it stops for a fraction of a second, the lone bass line starts up at a faster pace. Now how to I dance? Keep the slow pace? Hell No! The song builds and goes even faster and louder into a never ending outro. So, do we start fast dancing now? Oh, forget it! Extremely awkward. I don’t look cool at all! Just walk away. What a dumb song. …and that was my very first slow dance with a boy. Yes, the scars of embarrassment are still there.
I sense that you’re still bitter.
mmm
No. My last sentence was sarcasm. Just remembering my train of thought as the incident happened. Everything’s a bigger deal than it should be when you’re 14.
Phil “Scooter” Rizzuto. Former New York Yankee, then an announcer for many years, then he did commercials for The Money Store. Holy cow!
Led Zeppelin’s “The Song Remains The Same” has that extra long intro, goes into that different-tempo 1st (and, apparently, only) verse, then goes back to tempo 1 for the duration.
Paranoid Android - Radiohead
Daryl Hall’s “Babs and Babs”. Starts out as straight ahead pop, at the 3:35 mark switches to eerie soundscapes, and then a minute later jumps right back to the original arrangement! And just for good measure, starts getting spacey again for the last two minutes.