Mat Weddle's interpretation of "Hey Ya" as a folk song.
Though as we see in the "Rap and Hip-Hop" section of this page, "Hey Ya"'s pessimistic lyrics about love and relationships are actually more suited to this kind of format. It's the ORIGINAL that has a large deal of dissonance between the lyrics and the music.
"Merry Little Minuet," written by Sheldon Harnick and recorded by the Kingston Trio, is a cheerful little ditty about bad news from all over the world.
"Motherfucker Got Fucked Up", by Folk Uke, is a sweet, cheerful uke strum-along by two gentle-voiced girls... set to Gangsta Rap-styled lyrics of violence and profanity about hurting motherfuckers who get in their way.
Knock Me Up from the same duo is yet another cheerful ditty... about wanting to get pregnant via anonymous sex.
Kris Kristofferson. "Billy Dee" has an upbeat tune, but when you listen to the lyrics it's about a young man who gets lost in addiction and eventually O Ds.
Singer and guitar player Jim Croce has a few songs like this. One, "Salon and Saloon", should be happy and upbeat—it's a song about two old high school friends reuniting and chatting. Instead, it sounds like someone's entire family just died. Likewise, "Time in a Bottle" is a touching song about how much the singer cares for someone, yet it sounds wistful and bleak (perhaps justified, as the refrain says "But there never seems to be enough time to do the things you want to do once you find them").
Harvey Andrews' "Hey, Sandy", about Sandra Scheuer, has a fairly bouncy tune, to the point where at least one folk singer has demanded of an audience who seemed to be enjoying it "Do you actually know what this is about?"
"The Helicopter Song" a.k.a. "The Warder in the Joy" by the Wolfe Tones. If you just listen to the melody, you'd never realize it was about a prison break.
Hell, even if you listen, unless you know a little Irish English, you still might not realize it's about a prison break.
Canadian folk singer James Gordon gives us "Hockey Town", a jaunty pop-like number about a thirteen-year-old girl getting sexually assaulted by a seventeen-year-old member of her town's hockey team. Here's what happens next:
You know those tunes that just sound like they were written so that the American government could inspire patriotism? Set to one of those lyrics about how much he hates the place and wishes someone would "Bomb New Jersey".
We had to burn the city 'cause they wouldn't agree That things go better with democracy
Filker Tom Smith has the cheery tune "Walking Along the Beach..." The chorus starts, "I'm singin' a walking along the beach while you're slitting your wrists song." He lampshaded this in one live performance by commenting before the song, "Somehow, this has become a sing-along. Which means that one of us is really weird, folks!"
"For Lovin' Me" is a cheerful, happy song about how the main character has broken someone's heart and will break it again "someday when your poor heart is on the mend," plus has done the same to many others. And it goes on and on.
Tom Paxton's "Buy A Gun For Your Son" sounds rather upbeat and jaunty, and might be mistaken for pure jingoism... until you listen to the whole song.
With your picture on the wall He'll get that long-awaited call And press the firing buttons with a smile!
Pete Seeger's cover of Tom Paxton's "What Did You Learn In School Today?" is up there. A cheery little number that sounds exactly like a song a Kindergarten teacher would sing with their class, but is actually about all the lies children's heads are filled with in school as a form of indoctrination and an effort to raise them with naive, wide-eyed nationalism at the expense of the truth or compassion. Yikes.
"When in Rome" has elements of this as well, with the bits about the narrator brutally killing people generally being set to the most upbeat parts of the song. 
"Amie" by Damien Rice. A cheerful song with swooping orchestrals...about a man trying to convince a young girl to sit on his garden wall and read him a pornographic novel.
Also arguable for "Me My Yoke + I", which is a desperate, dark song about the joys of discovering masturbation.
"Shasta", a nice cheerful upbeat song about a girl taking a wonderfully pleasant roadtrip up I-5 in order to visit an abortion clinic. Okie-dokey...
It can be seen as a song about the beginning of a new life though, one can even interprertet it as a song about keeping the baby. The lyrics are fairly vague.
And on the other end of the spectrum is "Now Three" - By her own admission this song is an anxious and nervous happy song.
"Radio", in which a girl calmly and curiously imagines herself being mortally wounded in a suicide bombing in San Francisco.
Paul Kelly's rock song "How to Make Gravy" is a regretful song that takes the form of a letter written by a man who's in prison, to his brother. The narrator talks about the assorted members of his family, throws in tips for making the gravy that he normally makes, but it's completely heartbreaking when he talks about how it's nearly Christmas, and he won't be there to see his family- especially with the the line "Won't you kiss my kids on Christmas Day? Please don't let them cry for me."
"Jesuitmont" by the Celtic band Kornog, is a bouncy, energetic number - about an evil stepmother and her cook murdering her stepdaughter and baking her into a pie for her father to eat. (Why they choose to do so is never explained.)
Finnish folk-pop group Värttinä sometimes exemplify this trope, especially on their earlier albums, which feature dizzyingly chipper songs about unhappy marriages, villages full of idiots, and the general wretchedness of life.
"Matalii ja Mustii" is about a town where the girls are ugly, the boys are stupid, and the children are presumably below average. The lazy, experienced, alcoholic narrator is not impressed. This song was featured on the Arthur cartoon.
"Marilaulu" is about pouring boiling lead into gossiping old women's mouths, after cutting out their tongues.
"Barrett's Privateers", his most famous song, is an upbeat, a capella, easy to sing along with song about the eponymous crew's sole survivor describing the single battle the ship had and cursing everything for leaving him a broken man on the wrong side of the Atlantic.
"The Flowers of Bermuda" is a fast-tempo song about a shipwreck and the captain's Heroic Sacrifice.
"Flying", which sounds as majestic as the title suggests, is sung by a washed-up hockey star about how he's training another generation of kids to get used up and thrown aside by the sport.
"Canol Road" is the story of a man with cabin fever who murders a stranger in a bar and freezes to death while fleeing, interspersed with some Scenery Porn descriptions of the place where he died.
"Chane Ke Khet Mein" from the Bollywood movie Anjaam. The song lyrics are about a rape, but the song is very upbeat and sung at a wedding! It could be interpreted as being about a "rape fantasy" scenario or an instance of rough albeit consensual sex in a chickpea field, but still...
Bruce Cockburn's "Dancing In Paradise." Mellow, relaxing music, and the chorus is simply, "There's dancing in paradise . . . dancing in paradise..." However, Cockburn wrote the song in Jamaica and the verses are used to graphically describe the violence, poverty and corruption he saw there.
Zig-zagged in the 60s song "Red Rubber Ball", covered by various artists (written by Paul Simon and Bruce Woodley), has a very upbeat tune with lyrics about a guy who got ditched by his girl. The guy is fine about this ("The worst is over now, the morning sun is shining like a red rubber ball..."), but half of the song is a rant about how bad the girl treats him. ("The roller coaster ride we took is nearing to an end, I bought my ticket with my tears, that's all I'm gonna spend,")
Džamije Lete by Momčilo, sounds like a happy upbeat song...but the lyrics in Serbian are about blowing up mosques and terrifying acts of violence towards ethnic Bosnians.
The Tallest Man On Earth's "The Gardener" is a happy, catchy, cute tune about a paranoid man who kills any guy who could so much as bat an eyelid at his significant other (or who could possibly tell him about something he doesn't want to know about his SO, if you interpret it that way), and buries them in his garden to fertilize his plants. It's... jarring.
Andy Irvine's rendition of Marcus Turner's When the Boys Are On Parade is performed with Irvine's typical bouncy beat and rapid singing. Of such lyrics as :
Merely the whim or intuition of an elected politician
Makes a melee without conditions as the monster quits the cage
It's a machine that knows no quarter dealing death and sowing slaughter
Raping mothers, wives and daughters in an all consuming rage
Flogging Molly's songs often turn out this way. Almost all of them are in major keys with happy, fast-paced fiddle or pipe tunes as the melody. A recurring theme lyrically, on the other hand, is grief for lost youth, lost love and/or the generally crappy lot in life of the Irish. See "Light of a Fading Star"; "Tomorrow Comes a Day Too Soon"; "My Sweet Roisin Dubh"; "The Rare Ould Times"; "Tobacco Island"; and "What's Left of the Flag". "Screaming at the Wailing Wall" is one of their few songs not about these things, but remains a chipper-tuned downer.
The Pogues' songs usually have upbeat melodies with lyrics that are anything but upbeat.
Jonathan Coulton re-imagined "Baby Got Back" as a light-acoustic ballad and Destiny's Child's "Bills, Bills, Bills" as a bluegrass fusion number. The former is hilarious, the latter arguably works better than the original.
"Baby Got Back" still works after the humor of the dichotomy fades since his rendition makes the crass lyrics sound oddly touching.
"Re: Your Brains" is a song about a zombified office worker cheerfully trying to negotiate with his still-human co-worker ("All we want to do is eat your brains / We're not unreasonable, I mean, no one's gonna eat your eyes").
"Chiron Beta Prime" is a bouncy Christmas song set in the aftermath of a Robot War.
The lyrics of this one imply it may intentionally be invoking the trope in order to sneak it out past the Robot Overlords. Did I say overlords? I meant protectors.
"Shop Vac" is about post-suburban marital problems. Seriously.
"I Crush Everything", an extremely sad tune about the loneliness suffered by... a giant squid. Who hates dolphins.
Coulton also penned the lyrics and tune to "Still Alive", the ending song to the game Portal. It's a cheery little pop tune sung by the insane AI GLaDOS, with lyrics congratulating Chell in a very passive aggressive manner, as well as implying things are much, much worse on the outside of the Enrichment Center. ("While you're dying I'll be still alive / And when you're dead I will be still alive...")
Nor should you forget "The Future Soon", about someone dreaming of a future where he can build a robot army on a space station to conquer the earth and force the love of his life to be his bride...
A case of this done deliberately is "I Feel Fantastic". Coulton wrote the song after reading a Scientific American article about mood-altering medication. The song is a cheery tune about how great life is, but it quickly becomes clear the singer doesn't feel a genuine emotional state at any point in the song, instead letting medication control all of his moods.
Another rather deliberate instance is his song "Not About You", in which he insists that he's over his previous relationship and that he doesn't obsess over his ex, even though it's obviously not true.
Slashdot's unofficial anthem, "Code Monkey", is about a programmer who doesn't leave his crap job only to have a chance to see and chat with a secretary girl who won't even accept small gifts from him. It's also an another fine example of Coulton's love of shifting the focus back and forth to screw with people's minds.
"Blue Sunny Day" was written after Jonathan decided, just once, to make a song that was "kind of bouncy and happy". However, as he says, "once I had decided to use the phrase "blue sunny day," it was hard not to notice that the word "blue" can have another meaning. From there it's only a quick jump to vampire suicide." Notably, he tried hard not to make it about a sad vampire.
How about "Make You Cry?" If you don't listen to what he says, it sounds nice and peaceful...with lyrics like:
The love I hate The hate I need The pain that pulls me through I can't wait To watch you bleed When Your heart's broken too
"Betty and Me" is a very fast (almost frenetic) bluegrass sort of tune about how the narrator's relationship with his wife is getting better since they're having a baby, except for the many, many clues within the song that it's not his baby. Slightly subverted, since it's abundantly clear the singer is totally unaware of this and is genuinely happy about how "Betty says he'll be taller, and Betty says he'll be smarter, and Betty says that our baby will be better than me."
"Good Morning Tuscon" is about a news anchor who snaps and goes on a murderous rampage during a broadcast, then burns down the studio.
In fact, it's easier to just list the songs which are genuinely upbeat, so I will:
"You Ruined Everything," a song he wrote for his daughter, which, despite the title, is actually about how much he loves being a dad.
"Big Dick Farts A Polka," a song In The Style OfPaul and Storm, about Rich Wocjinski, "Big Dick to his friends, who despite his name is famous for what issues from his ass."
And "Dance, Soterios Johnson, Dance," about the secret double-life of his favorite radio host.
The Corrs have more than a few, including:
"Give Me A Reason", is about a relationship that was ended and the dumpee has no clue as to why.
"All In A Day", an intense song about how bad someone's life can get in one day.
Same thing goes for nearly every song on Thao Nguyen's latest album. With her gleeful, indie-folk style, loss and uncertainty never sounded so fun.
The 1967 song "Sunday Will Never Be The Same" by Spanky and Our Gang has a upbeat tune with beautiful harmonies, but the lyrics describe how breaking up with her lover has forever destroyed the singer's enjoyment of Sunday morning walks in the park.
"Wild World" by Cat Stevens (and by several artistes since) is a cheery little number about a parent warning his/her daughter, who's about to leave home, of all the dangers she faces out there.
Many interpret the folky harmonies of Stephen Stills' "Love the One You're With" to mean that it's a song about communal love and appreciation for what we have around us. However, lines like "Concentration slip away, cause your baby is so far away" and "There's a girl right next to you, and she's just waitin' for something to do" suggest a darker meaning, that the song would seem to be celebrating unfaithfulness.
Paul Simon did this in a few songs — "Mother and Child Reunion" is, depending on your perspective, a weirdly overwrought song about a custody battle or a gutwrenching story about a dying child (although Simon claims he wrote it about a chicken-and-egg dish he saw in a Chinese restaurant); "You Can Call Me Al" is essentially about loneliness and futile introspection with an anvilicious shot of "it could be a lot worse" in the third verse; "Me and Julio Down By The Schoolyard" is a rollicking happy tune about family rejection and unrest in poor neighborhoods.
The Mountain Goats are masters of this trope. "No Children" is kind of upbeat and perky — you could almost dance a jig to it — even though it has some of the nastiest, most spiteful lyrics ever committed to music. "Dance Music" is about stalking a girl and watching your parents' marriage fall apart, but the tune is similarly happy.
"You Will Burn" as recorded by Steeleye Span. The tone of the music calls to mind uplifting anthems such as "We Shall Overcome." The lyrics are about (and from the POV of) a group of people who break into your house, kidnap you, take you into the woods "where none will hear your cry," mutilate you, kill your children, raze your house, and burn you at the stake, all the while declaring that they're saving your soul.
Simon & Garfunkel's song "The Sun Is Burning". A sweet, melodic little piece, which is about a nuclear holocaust.
"I Am a Rock" sounds upbeat, but is about a recluse locking himself away. "The Boxer" is also up there in terms of this trope. The "Lai-la-lai!" in the chorus just adds to it. "Leaves That Are Green" is about the brevity of life and inevitability of death, and yet the actual music is lively and upbeat. (Blame the harpsichord)
Simon & Garfunkel are masters of this trope. "Richard Cory" is about a wealthy, successful businessman committing suicide. "A poem on an underground wall" is about someone writing the F word on a subway station wall. "A most peculiar man" is about yet another suicide, and ends with the lines "All the people said 'What a shame that he's dead, but wasn't he a most peculiar man?'" And yet, for some reason, everyone associates the duo with uncompromised, naive idealism.
Moxy Fruvous' "Drinking Song" sounds just like a drinking song (and a song about drinking) that you might find in an average pub, albeit with a tinge of the melancholy - until you listen to the lyrics and realize it's about how the singer's drinkin' buddy dies of alcohol poisoning - "He passed out on the sundeck that morning / quietly singing goodbye"
Similarly, their song "Independence Day" seems to be a sad breakup song, but there's a definite undercurrent of "Boy, I'm glad she's gone and I can breathe again."
One of the biggest crowd-pleasing numbers of Harry Chapin was "30,000 Pounds of Bananas", which tells, with jaunty high energy, the true story of a truck driver transporting the titular fruit to Scranton, Pennsylvania who crashes and dies horribly. He originally composed it at a more meditative piece, but when it emerged that the song was funny, he sped it up, and added a section where he riffed on his difficulties in writing a satisfactory end to the song. This didn't sit well with the trucker's widow, and it was agreed that Bananas would never be performed in Scranton, Pennsylvania.
Another Chapin tune, "Dreams Go By", is a bouncy-sounding number masking a bittersweet tale of two people whose childhood dreams are deferred and ultimately discarded by work, marriage, and family.
The Great Big Sea version of Captain Kidd, about a notorious English Scottish privateer, is very upbeat and cheerful. The song presents a (fictitious) account of the eponymous Kidd, wherein he is an Obviously Evil privateer captain who is a Jerkass to his dad, hates God, and kills people for fun. He tries to turn over a new leaf, but unfortunately, death does NOT equal redemption for Kidd, and he is hanged. However, in the end, he goes out with a smile, warning others to behave, "or you'll wind up just like me!". The song is a massive Ear Worm, by the way, as are a lot of GBS songs.
BTW, the song is Older Than They Think: GBS did a cover of a traditional Newfoundland folk song (as usual), which is purported to come from the time of Kidd himself.
Also performed by Great Big Sea, we have Excursion 'Round the Bay, which is a cheerful, bouncy song about a guy killing (accidentally or intentionally, it's not clear) his wife.
The Decemberists' "Culling of the Fold" starts out sounding like ragtime circus music—and then it starts telling you to cut people up. The chorus is about killing your lover to prevent overpopulation, or possibly to eat them.
"Scatterlings of Africa" is a cheerful-sounding and very upbeat piece of music... about the trials and difficulties of the uprooted and refugee populations of the eponymous continent.
"Walk You Home" by Passenger is an upbeat, bouncy tune that initially sounds like a song about the first flush of love with lyrics about accidentally touching hands at the coffee machine, but actually describes a stalker's infatuation with a work colleague that is out of his reach. It gets creepy when he starts sitting in a neighbor's tree with a pair of night vision binoculars, amongst other things.
"Jenny Again" by Tunng. Folksy guitar strumming in a minor, nonetheless peaceful, key. The song's lyrics are in the imperative - a man giving instructions to the friend of him and his girlfriend. Advice about what to do after said friend has murdered the man singing the song over the girlfriend, Jenny. The chorus tells the perpetrator not to worry "because no-one saw [the victim] fall". The serene style of the music certainly doesn't bring stabbing to the forefront of your mind.
Tunngs arguably most famous song, Bullets, is cheery and soft. Its lyrics feature such gems as "Our blood and guts are out", "We cut our fingers off to give ourselves a little extra insight" and "Carve angels on your eyes and all is undone".
The Gaelic song "Bean Phaidin" is a pretty fun song about petty jealousy, then you get to the fourth verse: "May you break your legs, Paidin's wife/ May you break your legs, your legs/ May you break your legs and your bones". Wait, what?
"Weile Weile Waile" by the Dubliners. Upbeat tune? Check. Happy children for the background vocals? Check. Infanticide and execution? Check.
It actually is a children's song from Dublin.
The song "The Irish Ballad" by Tom Lehrer. It's a song about a girl who kills her entire family—and then lets herself be arrested, "for lying she knew was a sin". And the tune is not dark or even melancholy, and it's completely matter-of-fact about the way she kills off her family members one-by-one.
Lehrer did this intentionally, as he felt it was almost stereotypical for traditional Irish ballads to be cheerful little ditties with ample amounts of nonsense lyrics ("rickety-tickety-tin") despite being about such pleasant topics as murder, suicide, and death in general.
"My Little Armalite" is actually quite cheerful sounding. Also, it's a song dedicated to an assault rifle, though the lyrics can be still humorous in the way they describe the tables turned on the British when the IRA is better armed.
The cheery and bouncy tune "Broad Black Brimmer" sounds light and happy. It even talks about a mother dressing her son in the clothes of the father. Very adorable. Except... the clothes are the uniform of the Irish Republican Army, and the boy's father survived the war against British imperialism, only to be killed in the resulting bitter civil war, and the boy is following in his father's footsteps to join the IRA and fight on for Irish independence.
Barleycorn's version of The Men Behind The Wire sounds peppy and cheerful in the melody, but the song is about racism, and political oppression (Not for them a judge and jury/Nor indeed a trial at all/But being Irish means you're guilty/So we're guilty one and all).
Generally invoked in-song Here's a Health to the Company. The singer and his friends are trying to have one joyous night before they go their separate ways, but the sad tone suggests they can't keep their minds off the fact that they might not ever see each other again.
"Do You Love an Apple?" appears to be a light-hearted ditty about two young people in love, but a closer look at the lyrics reveals that it's actually a cynical song about a woman lamenting her life going downhill because she got too attached to her drunken lout of a boyfriend and knows that she can't leave him. Even the title hints at this: it's an allusion to Eve eating the apple in the Garden of Eden—that is, mindlessly grabbing for something that seems sweet, but turns out to have unintended consequences.
Well Below the Valley is sung in an upbeat, sometimes seductive melody. The lyrics? Incest, rape, infanticide and the devil taking the victim to hell.
Clannad recorded a fast-paced, fairly cheerful version of "Two Sisters", a song about a young woman who murders her sister because she wants to marry her suitor.
Really, the fact we have a separate section for Irish folk says it all. Pretty much every upbeat Irish song is going to be this.
G. K. Chesterton: "The Irish are the race that God made mad, for all their wars are merry and all their songs are sad."