In WCW during the turn of the century, the term "foreign objects" (random objects usable as weapons in matches that don't allow them) was deemed racist by Turner Broadcasting standards and practices. They were renamed "international objects".
Also in WCW, when the Insane Clown Posse were working there, the "Violent" in Violent J's name was deemed unacceptable, causing the commentators to refer to him as Jake Jeckel.
When 'asshole' chants first became a thing in professional wrestling, Turner Broadcasting tried very hard at first to figure out a way of censoring the chants. Since there was no way to get the fans to stop and trying to mute the chants without muting whoever was talking was impractical, they gave up after a couple months.
During the early 2000s, UPN forced WWF to rename two of its wrestlers.
"Bad Ass" Billy Gunn's name was deemed problematic because of the gun reference. He went by Mr. Ass for a few years. (Yes, "gun" was deemed more offensive than "ass." Weird.)
Val Venis' name was deemed unacceptable because it rhymed with penis. One of his nicknames had been 'The Big Valboski', and that briefly became his real name.
Since action figures are marketed to kids, when Balls Mahoney was working for WWE, his action figure was named on the box as "B. Mahoney."
WWE's Divas Undressed special. Even though it aired during the graveyard block, they still spent a lot of time cutting away to judge reactions during particularly racy bits.
In one particularly disappointing example, the heel character of Muhammad Hassan was completely written out of storylines following the London terrorist bombing of the summer of 2005. This despite the character referring to himself as an Arab rather than a Moslem (although his ring entrance did make liberal use of Islamic imagery) and using a garrote made of piano wire as a heel weapon (a more stereotypically Italian way of murdering someone, and in fact Hassan's portrayer, Mark Copani, was partly Italian-American).