The expression is " I don't do __________"
It means the person won't do it or doesn't like it. In your example the person does not like to wait. It is similar to "I don't wait"
I don't do coffee. ( I won't drink coffee) (I don't like coffee)
She doesn't do windows. ( She doesn't clean windows)
We don't do purple. (We won't wear purple) (We don't like purple)