Gurindji woman Marii Oblelscuk, Ray Phillips, Conan Hammet and John Anstess are challenging health orders requiring most workers to be fully vaccinated.
The group says Chief Health Officer Hugh Heggie exceeded his authority and doesn't have the legal power to make the far-reaching directive.
Mr Burnside, a QC and author who unsuccessfully stood for the Australian Greens at the last federal election, told the Supreme Court in Darwin "this is not an anti-vax case".
"This is a question about statutory power ... Three of us at the bar table are all vaccinated," he said on Monday.
He said Dr Heggie had misused his extensive power and two of the four plaintiffs had already lost their jobs for refusing the jab.
"It does seem fairly tolerably clear that he has misapplied his power, that's the allegation," he said.
"The issue is plain and it is one that needs to be resolved ... the question is of public interest."
The workers are part of a broader group called United NT Businesses that are fighting the vaccination mandate, saying it is an undemocratic and draconian law that must be repealed.
It has more than 400 members but may be facing difficulties financing the battle, with Mr Burnside saying "we know we can raise $50,000" of the $300,000 required to bring it to trial later in the year.
The hearing for a protective costs order to cap the group's legal bill if they lose the case also heard the group had $200,000 in a trust to fund the fight.
"If an employee of the state is able to use his or her powers in a way that is not intended in the legislation then access to justice becomes vital," Mr Burnside said in support of the order.
NT Solicitor-General Nikolai Christrup SC opposed the cost cap, saying the plaintiffs were backed by a large group of businesses and the case was brought to court long after the mandate was launched.
The matter was postponed to allow Mr Burnside's team to amend the case paperwork to include Dr Heggie's recent health order requiring workers to have a third dose.
The territory's vaccination mandate was announced in mid-October.
It required tens of thousands of workers who interact with the public and vulnerable people to be fully vaccinated or face losing their jobs.
Outside of court, Mr Burnside said there was an important human rights issue about access to justice at the heart of the case.
He also said he opposed the anti-vaccination movement.
The matter will return to court on April 11, when a hearing date is expected to be fixed and Justice Sonia Brownhill will hand down her judgement over the cost order.
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