Be Careful What You Wish For (Situational Irony)
In Kansas, Dorothy sings a wistful song about the wonders that lie on the other side of a rainbow, only to find herself taking a ride inside a tornado to just such a magical otherworldly place. Once she does arrive "over the rainbow," she finds that she immediately wants to return home. This sudden change of heart is ironic, as it is a complete reversal of her initial desires in the film.
Professor Marvel is no psychic (Dramatic Irony)
When Dorothy visits Professor Marvel in his boxcar, she believes that he is a psychic, gifted with a connection to the world of the metaphysical. Going into his chambers, she asks him to look into his crystal ball and tell her fortune. He tells her to close her eyes, then looks through her bag for clues that might help him give a more convincing reading. Thus, the viewer knows that Professor Marvel is a hack, while Dorothy believes that he has an actual ability to prophesy the future, creating an instance of dramatic irony.
The characters already have what they're searching for (Situational Irony)
The Scarecrow constantly bemoans his lack of brains, yet proves to be the intellectual leader when it comes to planning and strategy. The Tin Man cries frequently and clearly has the capacity feel great emotion, in spite of his desire for a heart. The Lion, in spite of his fearful whimpers, has access to a well of bravery and courage, and yet feels that he lacks these things. Thus, there is an inherent irony in the fact that none of the characters can see that they possess the positive qualities they so feel they lack.
The Witch spies on Dorothy and her friends (Dramatic Irony)
As Dorothy makes her way down the yellow brick road, the Wicked Witch is able to watch her through her crystal ball at various points. The Witch knows Dorothy's every move and can send danger when necessary. While the viewer knows that the Witch can see everything Dorothy does and is sending trouble, Dorothy and her companions have no idea, which creates dramatic irony.
