In Avatar The Last Airbender this trope is played at least four times. First time is in The Winter Solstice Part 1 when Zuko saw Iroh get up from his hot springs and the object for censory was Zuko's hand blocking the camera. Later Iroh has a piece of cloth covering him. It the extras for that moment said "Earthbenders: 0; Naked Shackled Guy: 3". After the battle Zuko tells him to to put on some clothes. The second time is when Chong the nomad spots Sokka's underwear. Sokka uses Momo to cover himself up. The third time was in Aang's first nightmare in "Nightmares and Daydreams" when Aang goes into the Fire Lord's room and tells him he is gonna defeat him but we get this unexpected gem:
Ozai: Really? [Brings his face close to Aang.] How do you plan on doing that when you're not even wearing pants? [Cut to Aang's scared face, as he looks down to see that Ozai was telling the truth, and screams.] Aang: Oh no! [Aang quickly covers himself with a white Air Nomad shield pulled out of thin air. The camera zooms out to a larger shot of Aang shaking with embarrassment. Suddenly, giant eyes begin to appear all around him. The eye's close and disappear. Cut to shot of the giant Fire Lord laughing as Aang slowly backs out the door. The Fire Lord raises his arms to the ceiling, while surrounded by flames, laughing heartily.]
And in the same episode it is played again but reversed.
Ozai: Really? You're going to take me out? You're not even wearing pants! Aang: No, Fire Lord Ozai. You're not wearing any pants. [Ozai looks down and sees Aang is right. He quickly pulls over a shield with the Fire Nation emblem on it.] Ozai: No! My royal parts are showing. Noooo!
Belle: Are you wearing a grocery bag? Homer: I have misplaced my pants.
A similar incident happened on "Summer of 4 ft. 2", where Homer uses the welcome mat from Flanders' beach house as a bathing suit, and ends up getting arrested for indecent exposure when he steps outside.
From The Simpsons comics, Lisa had to wear a Titanic Squishee bucket in place of her standard clothes. Cue the squick.
On "Selma's Choice", Lisa had to wear a towel home after she got high off the hallucinogenic water on "The Little Land of Duff" ride and went skinny-dipping in Duff Gardens' Fermentarium.
King of the Hill: After getting his clothes blown off by a tornado, Hank Hill was once forced to make an on-the-spot choice between a potted cactus plant and the Texas flag for this purpose. He went with the cactus.
On the Mission Hill episode, "How to Get Head in Business Without Really Trying", Andy covers his crotch with a cactus plant as he's running from his boss (whom he thinks is out to get him for sleeping with his fiancée, but was really going to ask Andy if he wanted some sliced grapefruit for breakfast).
In the first season, Harold even had to do this with a pillow. His cabinmates hid all his clothes and moved his bed out to the dock (where all the girls were swimming the following morning) to get back at him for leaving his dirty underwear lying around.
Also in first season, Heather covers her breasts with her hands after a tree branch rips off her top.
The Planet Express crew is having a barbeque and Nibbler is begging for a burger, but no-one gives him one. Then Amy's falls out of her sandwich and lands in her cleavage. Seeing his chance Nibbler leaps onto her chest and devours it along with her bikini top, forcing her to cover up her breasts with the buns.
Done later in an Adam-And-Eve homage with what are noted to be "strangely sticky" leaves.
Wakfu. In the second season's Gobbowl arc, there is a magic potion allowing females to gender-bend into males, but which unfortunately cease to have effect if the subject says the word "Gobbowl". Thus, a Fanservice Extra trying to sneak in the stadium under the guise of a shirtless male has to cover her chest with an arm when she return to female form. Later, the same thing happens to a topless Amalia, who grabs a gobbowl shield to hide herself.
In the Codename: Kids Next Door episode "Operation: A.R.C.T.I.C.", after the KND are hit by a "Nude Torpedo", both Numbuh One and Numbuh Five manage to cover their bodies completely with both hands, thanks to the art style (giving them thin torsos and very large hands).
Taz-Mania: In "Kee-Wee Cornered", after Taz accidentally rips his own tail off, he covers his butt with a hat.
The Powerpuff Girls cover themselves with their hands as they return to the Professor after Nanobots eat all their clothing in "Nano of the North."
The Looney Tunes Show: Yosemite Sam covers himself with a cardboard box while fleeing naked from the FBI through the Spargle building in "Off-Duty Cop".
The Fairly OddParents frequently uses this trope especially in concert with another trope Naked People Are Funny. For example one episode has Cosmo and Wanda poofing Timmy out of the bathtub to his favorite comic store, but they forgot to dress him. As a result Timmy has to go through town covering up his "private parts" with his hands. He, also, uses a newspaper and his pink cap.
In a claymation short used before the second act of the episode "Bad Luck Bogus", Bogus does this with an item that he took out from a safe, after his pants get pulled off when they get snagged on the safe door.
Brattus does this with the bottom of his shirt after a cotton candy cone steals his pants in the first act of the episode "Waterboy Bogus".
In one episode of My Little Pony: Friendship Is Magic Rainbow Dash pulls Fluttershy's sheets off to get her out of bed, she gasps and covers her chest even though they usually don't wear clothes (spoiler: they're ponies!), Rainbow Dash gives her an annoyed look and Fluttershy grins sheepishly.
In the UPA cartoon version of The Emperor's New Clothes instead of being in his underwear after the swindlers give him his "new clothes" like in other versions he is naked, the first time his privates are covered by his mirror he is posing in front of, and as he's parading down the street to show off his new outfit they're covered by a flag, then by the townsfolk.
Strange Hill High: After Templeton has his trousers stolen in "The Lost and Found Boy", he hides in his locker covering his privates with a textbook.