Christian Metal in general. Of course, this is a religion whose most prominent symbol is the image of its God's mutilated corpse nailed to a cross, and some of whose followers ritually eat the flesh and drink the blood of said God in commemoration of His bursting forth from the grave, and a few groups see the metallic potential in that, but hearing super-positive lyrics (or 19th century hymns) beneath the brutal riffs and growled vocals is always a bit jarring.
One example of a different nature is Emery's song "Don't Bore Us Get to the Chorus." Its tone is much more upbeat than their other songs and has minimal amounts of screaming. It's about a serial rapist. That's arguably the point, though; the song is basically a deconstruction of pop culture's tendency to ignore disturbing lyrical content and go instead for a happy beat. The result is . . . memorable.
Christian rock band Newsboys has a song titled "Breakfast" a very cheery song with quirky lyrics...describing the death of a beloved member of a literal breakfast club. "Ah, rise up, Fruit Loop lovers, sing out Sweet and Low/With spoons held high we bid our brother Cheerio/When the toast is burned/And all the milk has turned/And Cap'n Crunch is waving farewell/When the big one finds you/May the song remind you/That they don't serve breakfast in Hell." The over-all message of the song isn't completely depressing - the Christian view that those who trust in God will be reunited in Heaven - but it's still a pretty cheery song for a song about death.
Relient K's song "Deathbed." The chorus, describing a man dying of cancer, is very somber ("I can feel the death on my sheets, covering me / I can't believe this is the end"), but the verses, reflecting on his life, are very upbeat, despite being about teenage alcoholism, parental abandonment, a shotgun wedding, divorce, more alcholism....
"(Near Sighted Girl with Approaching) Tidal Wave" (from Horrendous Disc) is a jaunty tune in the vein of the Beach Boys—with lyrics ending in "a watery grave".
"Ghost of the Heart" (from ¡Alarma!) has the creepiest music on the whole album, but its lyrics are about overcoming vanity and hatred.
"Hollow Man (Reprise)" (from Doppelgänger) promises that everything will get better, eventually—that "the form of every single grain will be restored in glory." It's set to creepy backing music (in fact, the backing music is the aforementioned "Ghost of the Heart" played backwards).
The entire album Vox Humana. With only one exception, the upbeat songs are about the failures of Western society, or about losing your soul to consumerism—while the somber songs are about meaningful relationships.
Five Iron Frenzy's "Blue Comb '78" puts humorous lyrics (eulogizing a comb that singer Reese Roper lost when he was five years old) to dramatically overwrought music (dramatically overwrought for a ska-punk band, at least). Subverted in that the song is actually a metaphor for his parent's divorce and his own lost innocence: Reese has stated that the lost comb incident was the last memory he has of his parents prior to the divorce. He chose to write indirectly to avoid falling into Wangst.
Five Iron Frenzy seems to be fond of this trope; consider the songs "Ugly Day" and "Where 0 Meets 15", both of which have relatively depressing lyrics set to upbeat, cheerful music.
The group Go Fish has "What Mary Didn't Know", an amazingly peppy song about a girl whom the narrator had the opportunity to lead to Christ but didn't before she died, and his angst over whether she's in Hell because of him.
"I knew the things to say, I knew the things to do I knew the people to know, but God, I didn't know you..." [Someday I'll answer for] What Mary didn't know was the answer I was holding I didn't think she'd change, so I never even tried How was I to know? I wish I would've told her Now I'll have to live in doubt, with what Mary didn't know...
Far too many hymns to list will have cheerful lyrics but slow, mournful melodies.
One egregious example is Christ lag in Todes Banden, (Christ Jesus Lay In Death's Strong Bands) by Martin Luther — the lyrics are joyful and celebratory about the resurrection of Jesus (it even has "therefore, let us joyful be" in the first verse!), but set to a dark, somber, minor-key melody more fit for a funeral dirge.
To be fair, the melody was written long before the invention of tonality, which governs key signatures and results in modern music's various emotional motifs.
J.S. Bach wrote a cantata based upon the melody and lyrics of this hymn, which means that the trope also applies to that cantata.
Another Bach cantata, Erfreut euch, ihr Herzen, is an inversion of the above. The cantata contains six movements, two of which are duets, where one duet part represents a doubter and the other represents a believer. The melody in both parts is light and happy, and the believer's lyrics celebrate Jesus's resurrection. However, the doubter's lyrics express sorrow that Jesus is dead and that "Death still holds Him in its bonds."
Conversely, the Values Dissonance between Christians and non-Christians means that a lot of non-Christians are disturbed by cheerful hymns about Armageddon like "Battle Hymn of the Republic."
No Hiding Place Down Here, made famous in Babylon 5 where several aliens and other non-Christians happily sang along, is a cheerful little ditty about how loosely defined "sinners" will be totally annihilated come Judgment Day. How cheerful.