On, in this case, means continue to do. - Carry on: continue what you are doing. - go on: continue what you are saying. Or continue going somewhere (go on without me) Dream on is a bit more difficult. It's usually said to someone that would like something to happen which is unlikely. For example: A: I can't wait to be a millionaire. B: Dream on. B thinks that it is unlikely that A will become a millionaire and that it is a more of a wish/dream/hope than reality. Up, in this case, means continue and/or finish. Use up: use all of it. Drink up: finish drinking the rest of your drink. Eat up: finish eating your meal. I've never see max up. This rule gets a bit more complicated with phrases like "hurry up", which means: move more quickly, or catch up with me. And "don't mess up", which means: don't get it wrong. These sorts of sayings are idioms, really. You will probably just have to learn them with context. They aren't often literal.

You are talking about phrasal verbs or another name is two part verbs. As Duncan says, there's no rule. You just have to learn each one as a new verb with a specific meaning. For example, sit up means to sit tall in your chair. Sit down doesn't mean to slouch in your chair.