The infamous Milli Vanilli episode, which got struck by this hard (read: heavily edited) upon syndication because of their 1990 scandal, which happened less than two weeks after the episode aired.
The episode becomes even more difficult to watch after Rob Pilatus' death from a drug overdose in 1998.
"Reptiles in the Rose Garden:" Mario accidentally sending the White House crashing on top of the Washington Monument became less funny when an earthquake affected said monument in 2011.
Also the episode with Dadzilla, who is introduced as a robot being used as a movie filmed in Hollywood. Eight years later, the American Godzilla movie was made.
Also in the Dadzilla episode, Big Mouth and Kootie Pie theorize that they are not really Bowser's children. Over twenty years later, Word of God revealed that the original seven Koopalings are no longer considered to be related to Bowser.
"Kootie Pie Rocks" is full of these. Specifically, the scene where they tell Kootie Pie they can't sing without the rest of the band. Their subsequent attempt to wing it (with Mario and crew) sends everyone running away screaming.
Several years after Milli Vanilli's guest appearance, Mario Party 4 had a minigame called "Blame It On the Crane".
Luigi refers to the tuxes he, Mario, and Toad are wearing as "these dumb penguin suits!"
In "Toddler Terrors of Time Travel", the Mario Bros., Toad, Koopa, and Kooky were all turned into toddlers before Yoshi's Island.
In "Mush-Rumors", the entire Mushroom Kingdom thought they were under an alien attack. Then came the Shroobs.
In "Super Koopa", we see King Koopa using the game's power-ups, including the Raccoon Suit. Over twenty years later, the False Bowsers in Super Mario 3D Land are depicted with raccoon tails, and in Super Mario 3D World, Bowser uses power-ups against Mario and friends.
The Koopa tanks and small airships only appeared in one episode each, for a short time each, and only the airships were relevant to the episode's plot. The battleships were left out completely, likely replaced with the Doomsub.
Lakitu made only one appearance in "Life's Ruff", tasked with distracting Luigi and King Windbag who have been turned into dogs from chasing Hip and Hop.
Just look at all the glaring animation and continuity errors. In one episode, Kootie Pie steals a fur scarf from a female mushroom which briefly turns into a sack of money when she throws it at Mario. It turns back into a scarf on the next shot.
Item blocks and clouds with eyes are only supposed to appear in the Mushroom Kingdom. That doesn't stop them from appearing on Earth in a few scenes, though.
Frank Sinatra was mentioned in "The Venice Menace" but referred to as Francis Sinatra.