Changing melody and tempo songs

Scenes From an Italian Restaurant - Billy Joel

Suite Judy Blue Eyes - Crosby Stills Nash

Uncle Albert / Admiral Halsey - Paul and Linda McCartney 

Tool "Lateralus"

Tool "Schism"

Paul McCartney " Uncle Albert / Admiral Halsey"

Wings "Band on the Run"

Beatles "Day in a Life"

The Who "Baba O'Riley"

Ike & Tina Turner "Proud Mary"

Dexy's Midnight Runners "Come on Eileen"

Yes "I've Seen All Good People"

Elton John "Burn Down The Mission"

I know what you mean by changing tempo (though a song can change from 4/4 time to 3/4 time and still remain at the same tempo. Tempo refers to beats per minute), but what do you mean by changing melody?

In the classic Pop song construct, there is the melody of the verses, a different melodic line in the choruses, and in many older songs (though not all) a third section, called the bridge (in the U.S.A.) or middle-8 (in England), again with its own melody

So any song with both verses (all songs) and choruses (many, but not all) there is a change in melody. Are you asking for songs with two different verse sections, each with its own melody, different from the other?

Interesting question, one I'll have to think about. Not that many, I have a feeling.

Almost my entire music collection, and the various genres I listen to, have regular shifting tempos, melodies, time signatures, chord progressions, etc, etc.

I listen to various subgenres of prog, jazz, and classical, all of which are well known for doing this.

The opening track from King Crimson's debut album, "21st Century Schizoid Man", after the intro vocal section, goes into a fast tempos jazz influenced frenetic instrumental section with different melodies than the opening and closing, followed by a kind of stuttering tempo, then, quite abruptly shifts back into the original tempo. All the while, shifting between 4/4 and 6/4.

Yes - Close to the Edge changes tempo, chords, time signatures so often, it's hard to keep up. It shits between 12/8/, 6/4, 4/4, at various times, and varies tempos from section to section. Also changes in melody, quiet and loud passages, mood changes.

Genesis - Super's Ready, changes tempos, and time signatures pretty often. There is even a section of the piece titled, "Apocalypse in 9/8". Each section of this song has related, but different melodies. 

Mahavishnu Orchestra has so many examples, it's kind of hard to list just one. But, "Dream" is as a good place to start. It starts with a slow tempo section that builds into a very fast section, that slows in tempo again. The melodies also change. I believe it shifts between 4/4 and 15/8. All done with terrifying levels of musicianship.

Return to Forever - Duel of the Jester and the Tyrant (pt 1 and 2) is loaded with tempo, melody changes.

And don't even get me started on the complexity of the tempo, time, and other changes in classical music of the later half of the 20th century, up through the present. 

More bands, and musicians known for tempo, melody, time and other changes:

Steve Coleman - contemporary jazz with lots of changes and complex shifting rhythms. 

Oregon - long existing jazz ensemble that pretty much defined modern chamber-jazz.

The Contortionist -  technical-metal band, with constant changes and complexity, and ridiculous levels of musicianship. 

PFM - Italian prog band, with a great deal of tempo and melody changes. Not to mention they could compose some drop dead gorgeous melodies.

ELP, Banco, del mutuo Soccorso, Dream Theater, Panzerballett, Frank Zappa, Gentle Giant, National Health, Happy the Man, Deus ex Machina, Magma... I could go on for pages...