Every good story needs an antagonist and the obnoxious voice is often the sure sign of one. This is one voice that just by hearing it enter the story gives you such an awareness of just what kind of person is entering the narrative.
There are many places where you will find an obnoxious voice in your story. It might be the know-it-all colleague or the opposition in any kind of team rivalry. The boss or leader of a new space the hero has entered in to. This is a voice of power and authority that says even in the way it speaks that "I know better!" or "I am better than you!" Disdain, superiority and class will give way to this voice.
The obnoxious voice generally gives the audience someone to root against. We cannot wait for them to get their comeuppance. They provide an extra reason for the hero to be successful. Thus it is valuable for drawing in the crowd who are watching to a form of group unity.
An obnoxious voice can manifest in a number of ways. There is usually a manner of condescension that accompanies a voice that comes across as sarcastic or annoying and unpleasant, perhaps grating. When class is involved it might be an over-enunciating of words as if to suggest they feel better in your mouth. An obnoxious voice is often used with questions about the supposed 'rights' of something that suggests the hearer is less worthy of an idea or place or position than the speaker.