I think one of the first and key pieces of irony that we are presented with in the novel is when the Dursley family try to do everything they possibly can to escape and run away from the letters Harry keeps on receiving from Hogwarts, but all it does is make things worse! Even when they completely de-camp and move to an isolated island they cannot stop Harry receiving the letter that tells him he has been selected to study at Hogwarts as a wizard.

If you want to think about other examples of irony in the story, you can think about the way that the author plants the red herring of the villain being Snape, when all along it is professor Quiddle. The fact that Harry insists that it is Snape all along is ironic given how close he is to working out the truth on a number of occasions.