If you are irreverent, you don’t show respect toward something, like your irreverent singing of your school's alma mater in a funny voice.
To be irreverent doesn’t necessarily mean you don't like something; it’s just that you don’t display the same honor for it that other people do. You might have a more light-hearted approach, poking fun of the way other people seem to almost blindly show their devotion. For example, when people at the Louvre are standing in hushed awe in front of the "Mona Lisa" it would be irreverent to holler "What's so great about that?"