In The Bible, Jesus would actually have been naked on the Cross; in most modern depictions, he's wearing a loincloth. Of course, the Bible never actually says whether he was naked or not; even the description of him being stripped (a key element of the Stations of the Cross) is not present in the text. Actually, this problem usually only comes up in Catholicism, since in Protestant churches it is taboo to show his body on the Cross at all.
Pretty much every piece of media that features the ancient Greek pantheon of gods will gloss over the fact that they're all siblings as not to Squick viewers out. Even the extremely gory God of War games glossed it over.
Of course, the fact that most of the gods were bi is always glossed over. Well, at least when it's the men, cuz, ya know.... can't freak out all the straight men reading these tales of badass muscled gods by revealing they liked some man ass too, now can we? Naturally, God Of War glosses over it too, most likely for the previous reasons: they wouldn't want to risk losing sales from their target audience by having Guy-On-Guy action in it.
Likewise, the gods' blatant adulteries are toned down. Zeus and Hera love their son Hercules, according to the Disneyversion. Now go and look up the original!
They also tend to ignore Athena's back story. Particularly the part where her dad ate her mom while she was still pregnant with Athena, who was then born out of her dad's forehead.
The Hades and Persephone myth is commonly known as "the Rape of Persephone". This title and the rape implied are often left out of retellings. Nor do they remind the reader that Persephone is his niece. While the story is called the "Rape of Persephone", by ancient Greece custom, abducting the bride from her home was part of the marriage ceremony, thus Hades legally married her. And marrying a niece kept family property in the family. On the plus side, they had the healthiest marriage of all the Gods that would require little additional censorship. However, it is also possible that this came from a different meaning of the word "rape". The Latin word raptio, often translated as rape, simply meant abduction of women. See http://en.wikipedia.org/wiki/The_Rape_of_the_Sabine_Women.
Most modern retellings of Theseus and the Minotaur fail to mention the fact that the Minotaur was the result of Minos' wife having sex with a bull. Or bring up the elaborate wooden cow "suit" that Daedalus built so they could perform the deed.
Many modern variants say that Aphrodite was born from seafoam. They omit the fact that the foam was actually from the sperm and blood spilled when Cronus castrated his father. And that's not to mention Hercules: The Legendary Journeys, where Hercules calls her "sister". Though some myths say that Aphrodite is the daughter of Zeus and Dione, making them half-siblings.
The Bowdlerisation of the gods' adulteries became a gag in the lostGilbert and Sullivan play Thespis, in which Daphne, playing Calliope, the Muse of Fame, uses a Bowdlerised classical dictionary to prove that Apollo is her husband:
Thespis: "Apollo was several times married, among others to Issa, Bolina, Coronis, Chymene, Cyrene, Chione, Acacallis, and Calliope." Daphne: And Calliope. Thespis (musing): Ha! I didn't know he was married to them. Daphne (severely): Sir! This is the Family Edition!
Just about every piece of ancient Anglo-Saxon literature we have falls to this. Seeing as the Anglo-Saxon's passed their stories on verbally, they would have all disappeared if it was not for the Christian monks who wanted the region converted. The monks recorded the Anglo-Saxon stories but inserted heavy religous bias. Two particularly egregious examples of this are in Beowulf and in the poem "The Seafarer". In the former, a hedonistic, bloodthirsty warrior with a God-complex is portrayed as a God fearing paladin of sorts. Meanwhile, the latter represents the last third of one of the most depressing ancient poems of all time (about a man's inability to find any contentment/his battle against his lot in life) as a praise song where the author revels in the glory of his creator. Never mind that Christian philosophy counteracts every major theme in these stories. Fate vs. free will. Materialism and living in the moment vs. spiritual rewards and living for an eternal future. Perhaps the monks thought that no-one else would notice.
Atum supposedly created Shu and Tefnut by ejaculating into his own mouth. Cleaner versions have had him simply spit on the ground and they were created from his saliva.
People are fairly familiar with the story of how Set murdered Osiris to get his throne only to be thwarted by Osiris' son Horus, but most people don't know how it was done. Set attempted to prove his worthiness before the other gods by anally raping Horus, but Horus reached between his legs and caught Set's semen, throwing it into the Nile. Horus proceeded to masturbate into a salad, which Set ate without knowing about the special sauce. When it came time for Set to prove his dominance over Horus, the gods commanded Set's semen to speak. When the voice came from the Nile, the gods then commanded Horus' semen to speak, and imagine Set's state of mind when his stomach started talking to him. That is how Horus avenged his father upon Set. The Egyptians were totally perverted. One version says that they had intercrural (thigh) sex and Set (or Seth) wanted Horus to catch the semen. He did and Horus' mom saw the mess, cut off her son's hands and threw them into the river. She then put her son's semen into Set's salad and the rest you know.
Plato was the Ur-Bowdleriser; in The Republic he explains how, in an ideal city, myths and epics would be edited to remove all mentions of gods and heroes doing bad or treacherous things, or even insulting each other, because gods are supposed to be unambiguously good (a very Platonic notion Homer would have had a hard time to comprehend) and that would be a bad example for the citizens. Knowing the nature of most Greek gods and heroes, he would have had a lot of work to say the least.
It goes back even further Plato. Xenophanes, a pre-Socratic philosopher, took great issue with Hesiod's Theogony (the poem which synthesised most of the myths about the Greek gods which which we're familiar today) for its characterisation of the gods as violent, cheating, debauched psychopaths, claiming such qualities were inappropriate for gods.
What gives "Rapunzel" away is changed from pregnancy to a Freudian Slip due to objections to her premarital sex with the prince; a pointless gesture, really, since she still bears his children before they meet again, let alone marry. Many modern versions omit the children altogether.
In Giambattista Basile's "Sun, Moon, and Talia", likely Charles Perrault's main source based on which he composed "Sleeping Beauty", the sleeping princess gets raped and impregnated by a king, and wakes up only after she has given birth to twins, and one of the babies suckles the cursed splinter of flax out of her finger. The mystery here is not why Perrault saw it fit to make some changes, but why Basile thought that something so jarring could be a charming fairy tale.
While Perrault's Sleeping Beauty is tamer and does not contain any rape, it still contains a cannibalistic ogre mother-in-law in the second part of the story, who almost eats the princess and her children. Victorian writers frequently cut out the second half of Perrault's version and ended it at the wedding of the prince and princess. Compare Perrault's original, uncensored text to this Bowdlerized version from the Victorian era. Nowadays, Perrault's version is left uncensored, though the Brothers Grimm's version (which always ended at the wedding) is much more likely to be used in today's fairy tale collections.
In Arthurian stories geared toward children, several major elements of the legend tend to be left out. In particular, the details of major character's parentage get cut out. Uther uses Merlin's magic to seduce Igraine as her dead husband and fathers Arthur. Arthur seduces/is seduced by his his half-sister Morgause and fathers Mordred. Elaine uses her magic to seduce Lancelot as Guinevere to father Galahad.
In the older version of "The Three Little Pigs", the big bad wolf torments and then eats the first two pigs, and is later boiled alive after he tries to climb down the third pig's chimney. Some modern re-tellings have the first two pigs managing to escape and making it to the brick house, where the smart pig manages to scare off the wolf by causing him a minor injury (a burnt hand, for example) or the wolf simply passes out from exhaustion brought on by the exertion of trying to blow the house down.
Victorian retellings of Donkeyskin frequently changed Donkeyskin's incestuous biological father to her adopted father or stepfather and her gold-pooping donkey had gold fall out of its ears instead.
Madame d'Aulnoy's The Yellow Dwarf ends unhappily, with the titular villain slaying the King of the Gold Mines, leading the protagonist Toutebelle to die of grief. The dwarf and the Fairy of the Desert get away with making the king and the princess suffer. When the story was performed in pantomime during the Victorian era, it was often given a happy ending, where the king slew the dwarf instead and married Toutebelle. (This also appeared in a number of children's editions of the story during the same time period.)
In Ferdowsi's revision of Ancient Persian tales, the The Shahnameh}} everytime the stunningly beautiful (and of course, virgin) daughter of a king falls in love with "Rostam" she begs him to marry her. In his BEDROOM, which she has sneaked into at MIDNIGHT.
Earlier versions of "Little Red Riding Hood" had more overt sexual themes, and were often warnings to young women not to trust young men who appear charming but are not. Red Riding Hood tended to be older, not a young child (in the Parrault version, she was an attractive teenager.) In these versions, the wolf eats the girl after she gets into bed with him, after being told to take her clothes off, or she escapes. The Brothers Grimm made her younger, changed the moral to don't talk to strangers and obey your parents, and removed the references to cannibalism and striptease. A lot of modern versions leave the part of the tale where Red Riding Hood chooses to go down the path of pins or the path of needles. While this seems innocent today, in some areas prostitutes used to advertise their profession by wearing needles on their sleeves.
In the original version of the Mahabharata, Princess Draupadi is married to all five of the Pandavas brothers. It is made clear in the narrative that although it's unusual (and kind of scandalous), there's no rule against it, and Draupadi is not the first or only woman to have multiple husbands. (It even mentions something along the lines of "If men are allowed to practice polygamy, no one should have a problem with a woman doing the same.") By the time the story made it over to Java, however, Islam had already taken hold there. And although Islam allows a man to have up to four wives as long as he can take care of them all equally, it specifically forbids a woman to have more than one husband at a time. So in the Javanese version of the story, Draupadi is married only to Yuddhisthra, although she has feelings for Arjuna (who was her favorite husband in the original.) The reason she collapses on the mountain climb to Heaven in the Javanese version is therefore her lust for a man who was a) not her husband and b) married to Subhadra. (In the original, the reason she collapses is that she loved Arjuna the most even though she was supposed to love all her husbands equally.)