These expressions are very commonly confused by Malaysians. I’ve even seen a sign in a bank with this mistake.
I’ll provide the definitions of each of these terms (from Wiktionary):
Fill in – (transitive) to complete a form or questionnaire with requested information.
Fill out – (transitive) to complete a form or questionnaire with requested information.
Fill up- 1. (chiefly of a fuel tank) to make full. 2. to become full
So we see that “fill in” and “fill out” mean to complete a questionnaire, survey or form with the necessary information. “Fill up” means to make something full, generally with a liquid. As mentioned by the dictionary, it is often used to refer to a car’s fuel tank. For example “I need to fill up my car”.
“Fill up” CANNOT be used to mean “complete a form”. This is wrong. Can a form hold liquid? (Well, possibly if you rolled it into a cone it could, but then it would be ruined). Since it cannot hold liquid, it is not appropriate to use “fill up”. The appropriate expression to use with a form is “fill in” or “fill out” (even though “in” and “out” are opposites, “fill in” and “fill out” both have the same meaning).
So please don’t ask anyone to “fill up a form” because they will not be able to do it. Ask them to “fill in” a form or “fill out” a form.