Look into the word irreverence and you'll see the verb revere, which means to admire, worship, or look up to. Irreverence, then, is the opposite: viewing something as not worthy of admiration.
The thing about irreverence is that you display it towards things that normally would call for reverence, such as a leader, a teacher, or a superstar. It's useless to show irreverence towards, say, a stray dog or a friend who's failed their driver's test the second time. Reserve irreverence for situations in which the object of your irreverence would normally inspire reverence. Addressing the president as "Dude" is the height of irreverence.