Malcolm Turnbull is standing by the deputy prime minister as parliament wraps up after a fortnight of sittings.
Mr Joyce has been referred to the court to test whether his New Zealand citizenship by descent disqualifies him from sitting in parliament under section 44 of the constitution.
"I think Barnaby Joyce should do the right thing, let the nation move on from this constitutional crisis that he and his colleagues have embroiled us in," Mr Shorten told reporters in Canberra on Thursday.
"He shouldn't be exercising responsibilities as a minister when we found out that he was a dual citizen."
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Dispute over Joyce citizenship, New Zealand role deepens
Mr Shorten said the government had failed to explain why another cabinet minister Matt Canavan, who has also been referred to the High Court over his eligibility, had stood aside while Mr Joyce had not.
Meanwhile, Treasurer Scott Morrison said Labor had been "sneaky" in working with NZ Labour colleagues to undermine Mr Joyce.
It was revealed this week an NZ Labour MP had asked a question in the New Zealand parliament about a scenario similar to that of Mr Joyce.
The query came after he discussed the citizenship issue with a staffer of Labor senator Penny Wong.
"Rather than come into the parliament and raise these questions, what they have done, in a very sneaky way, is run around over there in another country and try and dredge this stuff up," Mr Morrison said.
Labor MP Rob Mitchell brandished a tinfoil hat on Thursday morning, telling reporters Foreign Minister Julie Bishop - who championed the conspiracy argument in parliament - had lost the plot.
Media inquiries had already been under way when the NZ MP raised his query.
Another minister, Michael Keenan, rejected media reports he held dual nationality, insisting he renounced his British ties in 2004 before he entered parliament.
Fairfax Media had reported Mr Keenan may be a British citizen courtesy of his father Peter, who was born in England in 1943 and emigrated to Australia, where he married.
NZ Minister talks timeline
The timeline of events that led to Mr Joyce finding out he was a New Zealand citizen has been confirmed for the first time.
Internal Affairs Minister Peter Dunne said his department told him late on Thursday afternoon Mr Joyce's situation had been uncovered as a result of checks following Australian media inquiries.
"Given who he was, Foreign Affairs had been advised and had taken over the diplomatic management of it," he told RNZ.
"They in turn advised the High Commissioner in Canberra, who broke the delightful news to Mr Joyce... as I understand it, it was either late Thursday evening or Friday morning our time that Mr Joyce was informed."
New Zealand politician Peter Dunne addresses the media in 2013 in Wellington, New Zealand. Source: AP
Mr Dunne said he spent the weekend expecting the story to break at any moment in Australia, but that didn't happen until Mr Joyce announced it to federal parliament on Monday.
The minister reiterated the first inquiries came from Australian media.
"The facts show that inquiries into the possibility of there being people in the Australian parliament who might have been New Zealand citizens unwittingly were initiated by the Australian media."
Mr Dunne said Labour MP Chris Hipkins, who submitted written parliamentary questions to Mr Dunne about citizenship by descent, was "part of the process later on but he didn't initiate the whole process".
Mr Hipkins has been reprimanded by opposition party leader Jacinda Ardern for getting involved in the issue, which has caused fierce debate in Australia.
It's now known he submitted the questions after a call from a senior Australian Labor Party staffer - Marcus Ganley, who works for Opposition Labor foreign affairs spokesperson Senator Penny Wong.
Senator Wong has said Mr Ganley and Mr Hipkins had a discussion about the citizenship issue - which she wasn't aware of at the time - and thatthe New Zealand MP was not asked to put in parliamentary questions.
Mr Hipkins' involvement led Australian Foreign Minister Julie Bishop to say she would have a problem trusting a New Zealand Labour-led government.
Mr Joyce has renounced his New Zealand citizenship and his parliamentart fate will be decided by the Australian High Court.
Australia's constitution bars citizens of another country from standing for parliament.
