Songs that have false endings.

Exactly what I was going to say. Defines it.

Another: “I Just Dropped In (to See What Condition My Condition Was In”) by The First Edition.

Big crashing chord, then a few more sneaky notes, followed by another crashing chord.

“Free to Be You and Me.”

Strawberry Fields Forever

The Peppers’ version of Stevie Wonder’s “Higher Ground” tacks on a silly double-time coda.
The Stones’ “Stray Cat Blues” has a long, mellow, rather pointless coda, which they sensibly omitted when playing it live.
Eric Clapton went through a period in the 70s when he had trouble ending songs gracefully. I think it’s “Motherless Children” where the drummer adds a series of unnecessary cymbal crashes after the song is over.
Cat Stevens ends “I Think I See The Light” with a resolved major chord…in this case, ending on the unresolved chord would have sounded a lot better.

I haven’t heard it in a few years, but doesn’t “Carry On Wayward Son” by Kansas have at least two fake-out endings?

Chumbawamba’s “Tubthumping” (or is it Tubthumping’s “Chumbawamba”?) has an odd, Strawberry Fields-like instrumental coda.

There’s “Are You Experienced?” by the Jimi Hendrix Experience… IIRC, “I Don’t Live Today” also has at least one false ending.

Jethro Tull’s “For a Thousand Mothers” comes to a grand end before a set of cymbal taps ushers in a swirling instrumental outro.

At least two mentions of “Strawberry Fields Forever,” but nobody mentioned The Beatles’ “Hello, Goodbye” yet.

Not quite in the same category, though, as the note that precedes the “Hey la” coda decays, but ultimately goes right into the final part without pause. There’s no actual silence between the sections.

Plus, the less I think about “Hello Goodbye,” the better! One of a very small number of Beatles songs I actively dislike.

No Matter What by Badfinger

A more extreme, more awesome version: “Improper Dancing” by Electric Six. That link starts near the end of the song.At 2:50, the singer yells “Stop!” After the band goes silent for 2.5 beats, he yells “Continue!” The band plays the outro for about 23 more seconds.

I’ve always thought that Bonnie Tyler’s “Total Eclipse of the Heart” sounds like it’s ending several times before it actually does.

Then there’s this part of Rush’s “Time Stand Still” that sounds like it’s wrapping up, when there’s another verse on the way. It’s like the Springsteen clip in the OP.

John Entwistle did almost the opposite in I Wonder (which has a killer brass section, btw). The song abruptly stops right after the start of a verse.

Alanis Morisette in All I Really Want, though she does give it away slightly by saying “Here, can you handle this?”

Enya also has an unexpected pause in “Anywhere Is”.

Heard another one today. “Channel Z” by The B-52s.

Dudley Moore’s Beethoven parody from “Beyond the Fringe” has several points where it sounds like it’s about to end but continues. The first is at about 3:15. Dudley Moore Beethoven Sonata Parody - YouTube

Gee, no mention of one of the most famous musical pauses

ahhh… Bernadette !!!
      Wikipedia says

Absolutely. I opened the thread with SFF in mind. You speak true.

Another example might be “Final Solution” by Pere Ubu. The song comes to an abrupt halt precisely at its midpoint, after the line “Guitars trying to sound like a nuclear destruction.”

And then there’s Stoptime. Although this doesn’t actually fit the OP concept, because of the ostinato foot stomps, it does explore the concept of notes that aren’t played, in the spirit of the Tao Te Ching’s eleventh chapter on “the usefulness of what is not.” Likewise, Robert Fripp’s guitar solo on Bowie’s “Ashes to Ashes” makes excellent use of the notes that are not played.