The "walking/dancing on air" SPFX were done with a traveling matte. This is especially evident when Fred Astaire is dancing alone near the top of the monument. His cigarette smoke is visible only so long as it is over his body, vanishing immediately as it moves to the side.
During the "Currier and Ives" segment they are skating on a frozen pond in the "Winter" sequence. The refrigerant pipes for freezing the pond are clearly visible under the ice in several shots.
Mrs. Phineas Hill: All the cheques I've written to Charles, five of them. Supposedly for his pet charities. Well I'm through being charitable to everyone he pets.