I can remember several cutscenes.
Most of them have Axel in them and are from KH2.
For example Axel's first appearance in the middle of the Struggle Championship, the cutscenes before and after the fight between Axel and dual-wield Roxas, the cutscene at the Dark Depths after the huge Hollow Bastion battle (where Xemnas, Maleficent, Axel and Saïx appear), cutscenes in Betwixt and Between (Axel's death) and Sora finding Riku. I have memorized a few lines from other cutscenes too.
From KH1 I can only remember a few, one of them much better than the others; the cutscene before the fight with Riku/Ansem in front of the Hollow Bastion keyhole. (I had to watch it so many times, because I was stuck at that part once)