The scene at the end where Alice is running up a staircase, and she grabs a doorknob, only to have the door come to life and complain about her grabbing its nose. Mild and funny, right? HA. Try being six years old and having a crippling phobia of doors.
The drugged-up, blue caterpillar... They're just talking and then "I am exacicitually three inches HIGH, and THAT IS A VERY GOOD HEIGHT, INDEED!!!" *flashes psychedelic colors and explodes*
Alice: Well, I'm not used to it! And you needn't... SHOUT! (the way she says this causes the smoke from the explosion to disappear, revealing the remains of the caterpillar, now turned into a butterfly)
The caterpillar-turned-butterfly getting angry when explaining which sides of what exactly will make Alice grow taller or shorter: "THE MUSHROOM, OF COURSE!!"
Tweedledum and Tweedledee sneaking up on Alice, and the flowers turning on her and chasing her away.
That cage-bird-thing in the Tulgey Wood.
Of all the weirdness there, that's one of the weirdest. It (a cage with bird legs, neck, and head) keeps a couple of songbirds in it. When they get out (which naturally happens), it chases them down, snaps them up and swallows them alive - back down into the cage. And they're happy about it. More Nightmare Retardant then nightmare fuel, but it will still melt your brain if you think about it too long.
The umbrella birds which are revealed to be vultures.
David Hall's early concept art for Alice in Wonderland shown in various books and the DVD documentary Reflections On Alice. The Cheshire Cat has a mouth of pointy shark-like teeth and horrifically staring eyes, The baby's morph into the pig is horrific, the Mad Hatter and March Hare chase Alice with a large pair of scissors and a knife respectively, Alice is about to be decapitated by a grinding gear that may as well flay her, and it's all drawn in a horribly grotesquely realistic, ghastly style.
Another early concept featured an alternate version of the Cheshire Cat's famous disappearing act; instead of that funny unraveling thing he did in the finalized film, he takes his tail, sticks it in his mouth, starts eating himself and eventually, nothing is left but the grin.