Ring Around the Roses.
Not the song itself, God no, or the main lyrical performance, which was way better than I expected, but the backing vocals on the Partridge Family version of “We Gotta Get Outta This Place” are… well, they’re cheery. It’s like somebody decided to toss in a spare track
they had lying around from recording “Afternoon Delight”.
Still, if you were covered by The Partridge Family, Jello Biafra, and Arthur Brown I think you can say you have done well.
Also, “Fast Car”, which in my head is a response song to “Born to Run”.
I submit Tim Minchin’s “The Song for Phil Daoust”, when he goes after a reporter who gave him a bad review.
It’s been three years since you wrote it,
And time is very healing
But I still want to cut big chunks of flesh out of your stupid face
And make your children watch while I force you to eat them
Yeah, I wanna make your children watch you eat your own face meat!
Speaking of…
I’m not sure the tune is entirely “cheerful-sounding,” but the tune certainly doesn’t portend what the lyrics make clear: Timothy, by the Buoys.
I came in to mention Mack the Knife, but I see Hilarity N. Suze got it on Page 2. 
Harry Nilsson’s “You’re Breaking My Heart.” Cheery little tune with the lyrics:
“You’re breaking my heart
You’re tearing me apart
So fuck you.”
Nilsson also wrote the very cheery “I’d Rather Be Dead” and then did a video singing it with a chorus of senior citizens.
Yes, like that. It honestly took me years to actually look at the lyrics for “Carol of the Bells” to realize how very… cheerful the words were. The way it’s usually sung just does not lend to a cheery spirit (at least, nowhere near the level of the cheeriness of the lyrics).
Candy Everybody Wants by 10,000 Maniacs. Very upbeat music, complete with horn section, but lyrics that are an indictment of crass commercialism.
Sample lyrics:
“With A Gun” - more upbeat, downright perky music from the Minus Five:
*I’ll kick your sister’s ass
I’m going to take your brother’s face and smash it in the grass
It’s no wonder I am spiral-bound
'Cause I just want to
I just want to be around
Every day when I feel this way
I need somebody to say, “It’s okay”
Shoot some holes in my crow-black sky
Saying, “Life doesn’t really half-suck most of the time”
*
As the Clancy Brothers said between songs once: “Ahhh, Ireland. Land of Happy Wars and Sad Love Songs!”
First song where I ever stopped what I was doing, pulled out the lyrics (best thing about The Days Of Vinyl), and followed along. Perfect for this thread, it blew me away with phrases like
Four-poster/
Dull torpor/
Pulling downwards.
Country Joe and the Fish, “Feel Like I’m Fixin’ to Die Rag”, a Vietnam War protest song. Country Joe, by the way, was a Vietnam vet. Very peppy tune, with lyrics like “Ain’t no time to wonder why. WHOOPIE! We’re all gonna die!” Or “Be the first one your block to have your boy come home in a box.”
Oh man I thought of a great example of a terrible song. The Misfits’ “Last Caress” has terrible terrible horrifying lyrics, but really it’s an uptempo metal/punk song.
This video of Danzig performing has the song with lyrics.
She’s a Woman and Now He Is a Man– Husker Du
I know parodies don’t count, but…
Imagine You Were Mine–Mitch Benn
“Those Were the Good Old Days” from Damn Yankees (the musical).
“Cell Block Tango” from Chicago
The OP doesn’t specify pop music — there are loads of dark blues songs and ballads.
But two that come to mind are “The Band Played Waltzing Matilda” and “Fairytale of New York” by The Pogues.
Yeah, I came in to mention this one. A bit stalkery, and controlling.
'll always love you and make you happy
If you will only say the same
But if you leave me and love another
You’ll regret it all some day
Country Joe took the melody from Muskrat Ramble, a jazz piece that Kid Ory wrote in 1921. Ory’s daughter sued Country Joe for copyright infringement in 2001. The suit was dismissed because the Ory (and the Ory estate) waited too long to file it (Country Joe originally recorded “I Feel Like I’m Fixin’ to Die Rag” in 1965). There was also the little problem that Ory never registered the song with the copyright office, something that was necessary at the time.
The peppiness of the tune is part of what makes the Country Joe song work. It underlines the enthusiasm people were supposed to have for the war despite the reality of it.
Yeah, but these sound sad. The OP is asking for songs with cheery melodies but dark lyrics.
[Moderating]
A reminder, everyone, to only quote short excerpts of the lyrics, in accord with Fair Use of copyrighted material. If more is needed, link to the full lyrics elsewhere.
Goodbye Earl - Dixie Chicks
Possum Kingdom - Toadies. This song is not cheerful but it got a LOT darker for me when I realized it was from the perspective of a psycho luring his girlfriend/date to be murdered.
Battle of New Orleans - Johnny Horton. I have never wanted to be fighting in a war more in my life.