Louis Armstrong's version of "What A Wonderful World" is musical choice most often used to prove a cruel, mocking, and bitterly ironic counterpoint to the Crapsack World setting and the horrific action on screen. An effect helped by the fact that the song itself is haunting enough to be a Tear Jerkerdespite its upbeat lyrics. Furthermore, the song itself was written to try to distract people from what a crappy world was portrayed on the news at the time. That it's been used in the purpose of this trope is significant Irony.
Parodied in the Reduced Shakespeare Company's play, Completely Hollywood, where they end their major mash up of all movies with the Big Bad and hero confronting each other in slow motion. This very song is played as the hero dodges all of the villain's shots before killing him with one lucky one (as per cliched hollywood action movie ending).
Michael Moore's Bowling for Columbine, which uses both Louis Armstrong's in a sequence that portrays the dictatorships the US government installed or backed during the Cold War and ends with perfectly synchronized video of the attacks on the World Trade Center. It also uses Joey Ramone's version over the credits.
Played by Adrian Cronauer, as portrayed by Robin Williams, on his radio show in Good Morning Vietnam; the song runs over a montage of troops fighting in the field, officers at base looking reflective (and for once content with his music choice), and a village being carpet bombed.
And also played while a group of suspected Viet Cong are executed in an alley.
The Hitchhikers Guide To The Galaxy has a long history of abusing "What a Wonderful World." It replaced "Journey of the Sorcerer" as the music over the credits at the end of the first radio series, where Ford and Arthur, displaced in time, wonder at how beautiful the prehistoric Earth is, conscious that it'll be gone in two million years. The same scene appears in the TV adaptation, shifting into a readout on the Guide, floating through space, to explicitly remind us that in the show's "present", the Earth is destroyed.
A long-running Irish road safety ad had this as its theme. Apparently, no-one told Renault, who a few years later put an upbeat version as the theme of one of their ads...
Even anime gets in on the act here. The Whole Episode Flashback exploring Meia's tragic backstory in the first season of Vandread featured this song extensively. Then they subvert Soundtrack Dissonance by playing it completely straight at the end of the episode. Turns out, it really is A Wonderful World.
At Otakon 1995, someone combined this song with Fist of the North Star for comic effect. It was the definite CMOA of the music video contest.
Not even animated family films are immune. Madagascar uses it when Alex runs away after realizing that he might eat his friends. To hammer the point home, the others see a progression of cute little animals get eaten while the song plays. Wonderful world indeed.
Spitting Image used a parody version called "We've Ruined the World", sung by a weeping puppet caricature of Armstrong. The lyrics would fit the original tune, but for legal reasons the show was forced to use a different one.
Twelve Monkeys used the song extensively in what is almost a subversion of the trope, to highlight the beauty of the soon-to-be-ruined world.
The Mexican film Un Mundo Maravilloso (which appropriately translates as "A Wonderful World"), at the beginning, when we see the protagonist (a homeless, jobless hobo) wandering the streets at night seeking shelter from the rain and looking at the rich and middle class people in their homes from the outside. And the final shot in which after invading and taking over a middle class house with his buddies and family, the camera shows them celebrating while panning to the house owner's bodies in the yard.
Similarly, Israel Kamakawiwoʻole's "Over the Rainbow/What a Wonderful World", likely the second most popular recording of either, is almost exclusively used for a Tear Jerker. To be fair, the song itself is one for many people since the musician's early death.
There's a video on Youtube called "Satchmo's Lie", and plays "What a Wonderful World" to clips of nuclear tests.
Oliver Stone 's W. plays this song during a montage of the start of the Iraq War of 2003.
In the Season 4 premiere of Sons of Anarchy, the Alison Mosshart cover of the song is played by the band at Opie and Lyla's wedding, then it goes non-diegetic as the scene switches over to the rest of SAMCRO slaughtering The Mafiya.
Trailer for Russian 2013 movie Stalingrad utilizes altered and grim version of this song. Dark irony as it is.
Mild example in the JonTron episode "Home Alone Games", where Jon loudly sings part of it while cooking nuts so badly they explode. The explosions cut him off both times.