Accidental Innuendo: In "No Rabbit's a Fortress", Tigger tells Pooh and Piglet that he will lead them past the booby traps, and follows it with, "Tiggers is experts on trappin' boobies, y'know!"
Awesome Music / Ear Worm: The theme song. It was so kicking that all agree that the theme song is one of the best things to come out of the franchise. It's likely a whole generation of kids who grew up watching One Saturday Morning, the Pooh videos or other reruns have the theme drilled into their heads.
The closing credits instrumental version of theme song, particularly the extended version with humming and just a bit of vocals at the end.
Many of the music cues and background songs.
Bizarro Episode: This show had quite a few, most notably "The Piglet Who Would Be King" and "Cleanliness is Next to Impossible".
Nasty Jack, a literal horse, from "Paw and Order", to the point where he's playable in Mugen.
Kessie from "Find Her, Keep Her". Everyone remembers Kessie. This is probably why she was promoted to main character in the Book of Pooh.
Stan and Heff from "The Great Honey Pot Robbery" have quite a following, too, if DeviantArt is anything to go by.
"Funny Aneurysm" Moment: In "the Wishing Bear," the rest of the gang tries various ways to jog Pooh's memory when he forgets the rhyme needed to make a wish Christopher Robin's wishing star. The plot of the gang trying to help Pooh's memory was played much more seriously in Kingdom Hearts II when the Heartless tear pages of the Winnie-the-Pooh book out, which causes him to lose his entire memory.
The end of "Ground-Piglet Day" has Rabbit reveal that it's actually November 13th, which prompts Tigger to exclaim "November 13th!?" and gasp in horror, asking "Do you realize what this means?" He then goes on to say that it's "Porcupine Day", but after the November 13th of 2015 and the terrorist attacks on Paris, his intended over-the-top reaction to the date comes off as much more uncomfortable.
Pooh's cries to find the wishing star in "The Wishing Bear" become this when in Pooh's Grand Adventure, he similarly begs for the wishing star during his heart "Wherever You Are" - because he's "empty and [cold], and [his] heart's about to break". *sob*
Genius Bonus: In "No Rabbit's a Fortress", Gopher mentions that he has U235 in his arsenal of explosives.
Harsher in Hindsight: Several episodes deal with the theme of growing up (especially "Grown, But Not Forgotten"), and the theme would receive the most focus in the Darker and EdgierPooh's Grand Adventure: The Search for Christopher Robin. That film in general makes many of the moments in this series between Pooh and Christopher Robin much more bittersweet, as in hindsight, you now know that those precious moments are on borrowed time before Christopher Robin grows up and can't come to the Hundred Acre Wood regularly anymore.
Ho Yay: In "Un-Valentine's Day", Tigger is first to play the romantic lead in the valentine Rabbit is directing, but when Rabbit tells him he's got to kiss Kanga, Tigger is disgusted and refuses. Rabbit pushes for it, and Tigger agrees to do the kiss...but then immediately leans over and kisses Rabbit instead. Also, in "No Rabbit's a Fortress", Tigger lamented about how he hadn't given Rabbit a "good bouncin'" in days, saying, "I feel an empty spot in my life."
Memetic Mutation: From "Tigger Got Your Tongue": 'Looks like polar bears eating marshmallows in the snowstorm'note Tigger mistook a white piece of sheet for a painting that Piglet was supposedly doing