1. Sue accepted John’s invitation to the dance, but when Wayne asked her to go, she gave John the brush-off and told him she didn’t want to go with him.
2. The boss told me that he was busy right then and that I should come back later. I told him, “Don’t give me the brush-off. I deserve an honest answer from you. ”

The expression suggests the idea of brushing a small piece of lint or dirt from one’s clothing. It is usually used where one person is superior to (or thinks he is superior to) the other.