These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Foe Yay: There's plenty of this in "The Beauty of Kootie".
Fridge Logic: The episode "Do the Koopa" reveals that there is apparently a warp pipe that goes directly from the Mushroom Kingdom to Dark Land. This makes one question why, whenever Bowser kidnaps the Princess, does Mario need to go through eight worlds when he can just take this particular pipe.
Because they need two warp whistles to get to the pipe.
Fridge Horror: "7 Continents for 7 Koopas": Some of what Mario and Luigi do to chase Koopas away counts. Mario turns the Great Wall of China into a dragon and floods the Louvre, potentially destroying it while Luigi makes the penguins gigantic in Antarctica. Were Asia, Antarctica and Europe better off without the help of the Mario Bros.?
"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.
Hilarious in Hindsight: Again, Milli Vanilli's guest appearance. Specifically, the scene where they tell Cootie Pie they can't sing without the rest of the band. Their subsequent attempt to wing it (with Mario & crew) sends everyone running away screaming.
The episode where the Mario Bros., Toad, Koopa and Kooky were all turned into toddlers came out before Yoshi's Island did.
Milli Vanilli was a guest star in one episode. Come Mario Party 4 several years later, one minigame was called "Blame It On the Crane".
Also from the Milli Vanilli episode, Luigi refers to the tuxes he, Mario, and Toad are wearing as "these dumb penguin suits!"
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.
Idiot Plot: "Reptiles in the Rose Garden", "The Beauty of Kootie", "Never Koop a Koopa", "Reign Storm", "Tag Team Trouble", "Oh, Brother!", and "Mush-Rumors".
Jerk Ass Woobie: "Misadventures in Babysitting": Even though he's a brat and such a pain to Mario and Luigi, you can't help but feel bad for Junior when the Koopa Kids tricked him.
Memetic Mutation: "Permit me to introduce ya to... RACCOOOOOOOOOOOOOON KOOPA!!!!"
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.
They Just Didn't Care: 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.