The comic book writer and illustrator Don Rosa does it all the time: It's part of the appeal. There's hardly a panel without something funny happening in the background. If a painting (or any image of a character) is on-screen for more than one panel, it will invariably begin expressing emotion and reacting to what's going on in the foreground as if it were alive.
Not strange, since Rosa taught himself to draw so he could emulate Carl Barks, who did the same thing.
This is especially noticeable in Barks' Gyro Gearloose stories, where Gyro's little lightbulb Helper was always involved in some funny business in the background, sometimes even getting his own mini-plot line.
Though Rosa took it so far that once Barks commented that he was overusing the joke. Rosa and his fans respectfully disagreed.
Both funny and meaningful, in Rosa's early story, Last Sled to Dawson, Scrooge once again meets his old flame Goldie, who has since last time become owner of a hotel. As they have an awkward conversation, Donald is seen in the background, talking to the hotel attendant, and giving him a sly wink. If you're aware how much Rosa ships Scrooge and Goldie, you will realize that Don probably arranged for them to share a room without either of them knowing.
MAD doesn't have a page that isn't scrawled with miniature comics in the margins, footnotes, and assorted other wackiness. In fact, reading these can take up as much if not more time than reading the magazine proper (if such a word can even be said to apply to Mad).
The ur-artist of those margin comics is Sergio Aragonés.
FoxTrot often has small, easily missed gags in strips where the characters talk in front of paintings, posters, or photographs. A recurring gag is the object in question changing between panels (such as a person on a magazine cover being in a different pose). Likewise, whenever a character is reading a newspaper there will usually be a joke about a cartoonist in the headline.
Near the end of the Belgian comic Spirou and Fantasio in Moskow, one of the titular characters remarks that their speaking pet squirrel Spip has been very quiet, and the other says not to say anything, or Spip will start complaining again. Zoom in to the furious squirrel, who's been gagged with tape since an early kidnapping, which is clearly visible once you've been told. Interestingly enough, the squirrel stopped speaking in the next comics, until a change of artist.
A regular occurrence in the comic series Amelia Rules!; often accessible only to an older audience. Example: Amelia and friends attend Joe McCarthy Elementary School (motto: Weeding Out the Wrong Element Since 1952).
In Léonard le Génie, the cat and the mouse are usually engaged in their own activities when not commenting on the actions of the main characters. Especially in the older albums, these are small stories by themselves.
There's a scene in the original ElfQuest storyline where Cutter is planning to go off on his own and is telling the other Wolfriders he needs them to stay safe at home. He doesn't notice that behind his back Leetah is whispering something to Skywise (ie she's persuading him to accompany Cutter).
Chilean comic book Condorito is famous for this, with a crocodile slowly crawling inside someone's house, a sleepwalker about to fall down a open manhole or a soccer player's picture kicking it's ball out of the frame and then trying to recover it being the most famous ones.
Ibañez, author of the Spanish Mortadelo y Filemón comic books series, absolutely loves to do this. It is not a rare occurrence to spot a random pedestrian with his nose coming out the back of his head, pigeons with horse-like faces, the bare feet of someone sleeping in an unorthodox place like a folder cabinet or a trash bin, mouses chasing cats chasing dogs, anthropomorphic buildings, cars parked upside down... The list goes on.
The UK comic artist Tom Paterson (Calamity James in The Beano; Sweeney Toddler in Whizzer And Chips) is extremely fond of this, to the extent that sometimes the main action of the strip almost gets crowded out of the panels. (His "little squelchy thingies" in Calamity James eventually got their own full page spotters' guide◊).
Chick Tracts use these quite frequently, usually in the recurring appearance of Fang the dog. Frequently during the obligatory one-to-one preaching scenes, Chick will pull these into the foreground so he's not just drawing talking heads in every panel.
Done in Calvin and Hobbes when Calvin's Mom says that she hasn't seen Calvin for about fifteen minutes. We then see out the window while she's reading the newspaper, where the family car is rolling into the road and Calvin and Hobbes are chasing it. In the last panel, she says that it probably means he's getting in trouble.
Top 10's setting of a city where every last person has superpowers and an outfit to match means there's probably more panels with something odd in the background than not.
In a Justice League issue from the 1990s, the League has just dealt with an attack on a city by evil versions of themselves. After the villains are taken care of, an angry citizen chastises them for not stopping them sooner, never mind that the League was missing several key members during the battle and that given the power levels of the villains, the casualties could have been far worse. Green Lantern starts to argue with the man until Superman says that he'll handle it. The focus then shifts to the rest of the League discussing the threat, but Superman and the man are still seen in the background. In the first panel, the man is shouting at Superman. In the second, Superman is calmly speaking to the man. In the third, Superman and the man are respectfully shaking hands.
The illustrations in Graham Oakley's Church Mice books are nothing but this.
Used frequently by the comic artist George Herriman, often as Running Gags. A recurring Funny Background Event in his comic The Dingbat Familyeventually led to his most famous creation, Krazy Kat.
In one Krazy Kat strip, there is a Spanish-language poster that changes with every frame (and changes fence side for one frame), but is always a word or phrase in Spanish with an image that matches the text. Special props to the one on Ignatz's side of the fence that reads "Ratón Muerto", Dead Mouse.
Used in What's New? With Phil And Dixie, with things like people in dice costumes walking behind the title characters, or little gag scenes going on in the back ranks of crowds.
In the short-lived Emma Frost comic series, Emma uses her psychic powers to manipulate a student who is trying to get a teacher (who is also Emma's Love Interest) fired. While doing so, Emma acts out the things she's making the girl say and do, resulting in Emma standing in front of a building, ranting and flailing. In one panel, another student can be seen giving her a very weird look.
Astérix features a fair number of humorous gags in the background. Usually they involve animals being involved in some sort of minor gag, but oftentimes they're nameless people involved in an unspoken Orphaned Punchline. The familiar 'are you crazy' gesture appears in quite a few of these.
One time, Power Girl and Terra duck into an alley to change into their costumes, with Power Girl lecturing Terra on how she should have Dressed in Layers. A bum in the alley enjoys the free fan service.
In the first splash page, a poor background pony falls victim to an Anvil On Head.
Same issue, look closely during the Changelings battle when Rainbow Dash and Fluttershy are talking and you'll notice Spike running away from a Changeling he just set on fire.
While Spike is horking up Queen Chrysalis' message after the changeling battle, Pinkie Pie can be seen re-inflating her mane to its normal wild curliness (much to the chagrin of Rarity, who had styled it during the battle).
Paul Sample's Ogri cartoons have entire little subplots taking place as Funny Background Events. For example, a dog is looking down a drain; then the drain eats the dog; then the drain belches; then it spits the bones out. Or the main sequence of the strip involves a bike cop stopping Malcolm and rabbiting on endlessly as he pulls charge after charge out of the "Police Officer's Handbook of Fantasy Charges" to charge Malcolm with, while in the background Ogri is taking the cop's bike apart and selling the bits to random passers-by; when the cop is finally finished and turns back to his bike, Ogri has disappeared and nothing is left of the bike either.
Allison Bechdel's Dykes To Watch Out For contains lots of background goodies, such as a sign reading "Try our gluten-free, lactose-free, low-fat white pizza!"