As a New Zealand citizen, do I have to pay a fee for high school education in Australia?

Not in a State school but you will pay for school uniforms, levy fees and you always seem to be paying for something eg: excursions etc.. They do have a good system for text books in most Brisbane high schools where you hire the books and the govt. subsidy for text books which you receive is passed on to the school. High school usually have uniform shops, where you can get good 2nd hand uniforms or buy new ones. Of private education you pay yearly fees.
As well as everything else. else.

No you don't pay unless you have chosen a private school . State schools are definately worse here than in New Zealand .