The Adventures of Super Mario Bros. 3 contains examples of:
Absentee Actor: Between four heroes, one Big Bad and seven Koopa Kids, there were bound to be some absences. "Life's Ruff" is a special case. Most of the regulars are absent, but so is Mario himself—making for the only episode of the three shows that he didn't appear in. (Incidentally, this means Luigi is the only character to appear in every episode of every show.)
Adventure Towns: Many episodes take place on a location somewhere in the Mushroom Kingdom or Earth.
Affably Evil: King Koopa, in contrast to his Faux Affably Evil nature in Super Show and his Card-Carrying Villain status, he actually does genuinely care about the Koopa Kids. He especially loves for Kootie Pie to the point where he'd conquer America for her if she asked for it, just to make her happy, though even he knows how to discipline them when needed.
All Your Powers Combined: In "Super Koopa", Koopa is able to combine the Mario's powerups, something they can't do. At one point, he uses Raccoon, Fire, and Kuribo's Shoe all at once.
The writer's bible also gives some interesting character tidbits you might not know otherwise:
Before he was the Princess' servant, Toad was a pizza boy and encyclopedia salesman, after that he was the personal chef of the Mushroom Castle.
Bully Koopa loves to listen to 50s greaser music in Nintendo style.
Kooky Von Koopa is a graduate of Frankenstein University. He's also the one who built the Doom Ship and is responsible for the creation many of the monsters and creatures seen in the series, also his favorite foods are candy and fried foods.
Cheatsy Koopa's hobby is counterfeiting gold coins.
Hip and Hop have the most powerful magic wands out of all the Koopa Kids.
Appliance Defenestration: Kootie Pie does this to her birthday presents and cake at the beginning of "Reptiles in the Rose Garden".
Arson, Murder, and Jaywalking: In "Crimes R Us" the Koopa Kids steal the following from the Mushroom people: 522 wallets, 391 purses, 912 gold coins and one yo-yo.
As Long as It Sounds Foreign: Parodied in "The Venice Menace" in which the Italian language is made up of nothing but names of famous Italians and Italian-Americans. Hilariously, one of the names uttered is "Lou Albano", AKA Mario's old actor/voice actor (he and Danny Wells were replaced by the time of this cartoon).
Badbutt: Bully Koopa, who's portrayed as a kid-friendly gangster type.
Bag of Kidnapping: In a rare villain example, this happens to the Koopa Kids in "Crimes R Us". Crimewave Clyde and the Mario Bros. throw them in a dollar-sign bag bound and gagged and sends it to Koopa in the Doomship.
Birthday Episode: "Reptiles in the Rose Garden" takes place on Kootie Pie's sixteenth birthday.
Extra Hindsight Hilarity points for the fact that when Milli Vanilli point out they can't be forced to do a song without their backup singers. Taken further in that when Kootie Pie gives up on having them as her captives, she laments that their music was too beautiful to last!
Comically Missing the Point: In "Do the Koopa", Koopa is using the magic music box to force everyone to dance, including his traitorous kids.
Deus ex Machina: The wands were used like this a couple times, but this was averted in "Recycled Koopa", where magic didn't fix everything.
Disability Immunity: Bowser has a cold in "Do the Koopa", rendering him hard of hearing. This means he can't hear the Doom Dancer Music Box's music, and therefore isn't affected by it.
Divide and Conquer: Said word-for-word in "True Colors" as the Koopas plan to turn the Mushroom people against each other.
Dub Name Change: The writers made up their own names for the Koopalings, called the Koopa Kids in the actual show, before the English version of The Manual was final. That, and some of the references Nintendo ended up using would likely go over most kids' heads anyway.
Easily Conquered World: Earth itself. Several episodes feature King Koopa taking over a country or a city with minimum effort. One even has each of the Koopa Kids taking over the continents, with nothing more than their magic wands.
Episode Title Card: Done in the style of the title screen for Super Mario Bros. 3 (a curtain raising), naturally (except "Kootie Pie Rocks").
Fanservice: Princess Toadstool wears a swimsuit in more than one episode. Sure, it started as a Beach Episode, but at one point, she gets captured wearing it and spends the whole rest of the episode clad as such.
There's also her Totally Radical '80s outfits in the episode "Kootie Pie Rocks".
Fantastic Racism: Exploited by the Koopas in "True Colors", who paint the townsfolk red and blue and spark riots between them.
Mario: You're so stubborn now that you're red, Luigi! Luigi: Yeah? Well, you're a blue know-it-all!
Foot Focus: A barefoot Princess Toadstool in "Reign Storm".
Heel-Face Turn: During the episode "Crimes R Us", King Bowser Koopa abducted Crimewave Clyde out of prison to train his kids how to become hardcore criminals. They became too much for Clyde and after the Koopa Kids double crossed him at the Mushroom Kingdom Treasury, he switched sides with the Marios and gave Bowser and his kids a lesson in payback. Though Clyde did return to prison, he was more happy to 12,423-year prison sentence with no parole than to spend another day with the Koopas.
King Windbag of Ice Land also had one in "Life's Ruff". Princess Toadstool accused him of being a bully and sent Luigi to his castle. He witnessed the king mistreating his servant and threatening him with his wand. He and Luigi were turned into dogs by Hip and Hop and were eventually sent to the pound by a dogcatcher. King Windbag was cajoled by Luigi to work with the other trapped dogs and together they escape. After turning himself and Luigi back to humans, he arrived back at Ice Land with a warm heart and prepared a feast for his servant.
Help, I'm Stuck!: When the crank breaks off the Doom Dancer Music Box, Koopa gets his claw caught in the hole. It falls off the Doomship and gets destroyed when Cheatsy, Big Mouth and Bully pry it off.
Idiot Ball: Princess Toadstool picks it up in "Never Koop A Koopa." When King Koopa pretends to give up and hands her the key to Castle Koopa, she decides that she'll turn the castle into a retirement home for elderly mushrooms and runs off to check it out.
Leitmotif: Since the show re-arranges most of the music from the game, the tunes are indicative of the character or type of environment that is currently on-screen. Examples: Water Land's theme playing any time there's a swimming scene; Dark Land's theme playing whenever King Koopa or his fortress is on-screen; the airship theme whenever Koopa's airship appears. And appropriately, most episodes ended with the credits theme from the game.
Lolicon: Subverted in "The Beauty of Kootie": Mario uses Kootie's wand to transform Luigi into "Casanova Koopa". Kootie Pie immediately falls for him and kisses him on the lips passionately, which disgusts Luigi himself while in disguise. After Kootie accidentally changes him back (to say she overreacted would be a HUGE understatement), she spits out Luigi's germs.
MacGuffin: The wand in "A Toadally Magical Adventure."
Magic Music: The Doom Dancer Music Box made anyone who could hear it dance uncontrollably. This became a plot point as Koopa, who had a cold, couldn't hear it and thus wasn't affected when Bully tried it on him.
Make Up Is Evil: In "Reign Storm" the Koopas create an evil robot version of Princess Toadstool. Who wears really dark lipstick.
Mama Bear: In "Mind Your Mummy Mommy, Mario," Queen Mushroomkhamen becomes enraged when her son's sarcophagus is stolen.
Mermaid Problem: In one episode Mario was rescued by a mermaid named Holly Mackerel, who fell in love with him. Complicating matters was the fact that she was a reverse mermaid, and that she saw him in his Frog Suit, which made her think he was a frog.
Mind Control Music: The Doom Dancer Music Box makes anyone who could hear it dance uncontrollably. It can also be used to march others away, and if cranked slowly, makes them slow down and stop moving.
Mooning: Probably just a goof, but Luigi does this to Koopa in "Misadventure of Mighty Plumber".
He also floods the Louvre to fix the artwork that Kootie Pie ruined.
Luigi makes the penguins gigantic in response to Big Mouth's Abominable Snowkoopa.
Near Villain Victory: Koopa came THIS close to winning in "Never Koop A Koopa". He had everyone captured and at the mercy of Bob-Ombs...and lost cause Cheatsy decided to cheat him.
He came as close in "Crimes R Us". The entire Mushroom Kingdom Treasury could have been Koopa's for the taking had it not been for Crimewave Clyde foiling the Koopa Kids' plan to destroy him and his subsequent Heel-Face Turn.
Koopa wasn't immune from these either. His plan in "Sneaky Lying Cheating Giant Ninja Koopas" went sour in a heartbeat after he accidentally turned Mario into a giant.
Not in Kansas Anymore: The human father from "Mush-Rumors" utters this when they drive through the Mushroom Kingdom.
Off Model: As with the Super Show, Mario and Luigi still appear off model with their overalls and shirts having the colors swapped, despite that the appearance they have today was used in the game this cartoon was based on.
Our Mermaids Are Different: In "The Ugly Mermaid" Mario met a tribe of mermaids who are fishy on top but have human legs. They can also only breathe water and need water-filled bubble helmets out of it. For some reason they live in an air-filled dome, yet King Koopa flooding it's treated like a bad thing.
It seems they were intentionally mocking this trope, since when Koopa sends Kooky to Hawaii to make sure the Princess stays there, all he gives him to fit in with the humans is a lei to wear around his neck and it works.
King Koopa and Kooky dress up in plumber's clothes in "Toddler Terrors of Time Travel" and proceed to clog an apartment's drainpipes.
This doesn't work for the good guys. In one episode the Mario Bros. try to disguise themselves as sledge brothers to stow away on the Doomship. Luigi is worried that it won't work, Mario tries to assure him that he barely recognizes him. But once they get aboard, an actual sledge brother deadpans "Puttin' on a little weight, aren'tcha Mario?"
Double Subverted with Kootie Pie's human transformation in "The Beauty of Kootie". Though Cheatsy pulled off a believable disguise, he forgot to remove her tail which Luigi and Princess noticed but not Mario.
Politicians Kiss Babies: Bowser does this in an episode. He's only able to go through with it thanks to a "Nice Guy Potion" cooked up by Cheatsy and Kooky.
Portrait Painting Peephole: Demonstrated in "Never Koop a Koopa" when Koopa watches Princess badmouth him through the eyes of his own portrait.
Mario: Well, I wanna listen to music! This record's hot!
Luigi:(takes the record and tosses it in the fireplace) There, now it's even HOTTER!!
Produce Pelting: Bystanders throw various items at Kootie Pie and Big Mouth during their number, "Stomp My Name on the Walk of Fame" in "Dadzilla", burying them in trash by the end of the performance. They also do this to Madzilla.
Rogues Gallery: The Koopa Kids rotated as the main villains of each episode but only a few ("Crimes R Us", "7 Continents for 7 Koopas" and "Mush-Rumors") had all seven working together.
Role Reprisal: While Mario, Luigi and Princess Toadstool got all new voice actors, Bowser and Toad maintain their own voice actors (Harvey Atkins for Bowser and John Stocker for Toad).
Shown Their Work: This show was the closest adaptation of the games, even featuring P-Wings and the Kuribo's Shoe in an episode, as well as the ruler of Desert Land. Ice Land's ruler is also featured but is given a different appearance from the game.
The music box in "Do the Koopa" may reference a similar item from the game.
To add to that, the Paper-Thin Disguise Mario and Lugi wear in "Reptiles in the Rose Garden" are reminiscent of the Hammer Suits.
A Storm Is Coming: In "Oh, Brother!" Mario and Luigi's argument hits its peak as it starts to rain.
Super Power Meltdown: Toad and Princess Toadstool use a filled-to-the-brim chest of power-ups that Toad well-stocked in "Super Koopa" to keep Mario and Luigi powered up each time King Koopa de-powers them, though they sometimes cause Koopa to be de-powered. Both sides use their power sources (a pendant for King Koopa, the chest for Mario and Luigi) to keep the fight going. The constant stress of this, however, causes King Koopa's pendant to short-circuit, and we see him turn into many powered-up King-Koopa versionsbefore he finally explodes into his normal de-powered self, the pendant useless as he's sent through a manhole into the Mushroom Kingdom again.
Big Mouth: It's curtains for you creeps! You're history! You've had it! You're done for! It's all over! We win! Time's up! Your clock is cleaned! The sun has set! The fat lady has sung! The cheese has molded! The Super Mario Bros. are no more!
Water Is Air: In "The Ugly Mermaid", King Koopa tries to make a kingdom of fish surrender to him by flooding their air-filled dome with water, to which they react negatively for some reason.
The merpeople even have to wear water-filled globes over their heads so they can breathe while in the city, Seriously, what?
What Happened to the Mouse?: In "The Beauty of Kootie", Kootie Pie locks Princess, Toad, Desert Land's sultan and one of his guards in a dungeon. When she releases them, the guard is nowhere to be seen.