Former Ngapuhi Runanga chairman Rudy Taylor, on behalf of the Dunns, told Judge De Ridder the home was considered by the family as their urban marae with afterbirths from mokopuna buried there and family members lying in state there.
Mr Taylor said the Dunns wanted to buy the home from HNZ and wanted the stay to allow this proposal to be explored.
But Siobhan Buckley, on behalf of HNZ, said the policies had been enforced correctly and the tribunal had backed this up by issuing an eviction notice.
Ms Buckley said HNZ had a waiting list of tenants and its policy was that once a tenant died, there was no right of succession for a family, regardless of how long they had lived at the property.
Judge De Ridder said there was an incredibly strong connection between the Dunns and the property and they considered it their turangawaewae.
"There's a very deep and significant wairua between the family and the property. HNZ applied for possession, which has been granted, and the family wants to stay in the house," the judge said.
However, he said, the Tenancy Act was paramount and HNZ had followed its policies and obligations correctly.
"When it comes to the question of standing ... the tenancy was in the sole name of Rina Dunn. When it comes to the appeal ... in my view there's what might be insurmountable difficulties. The [chances of winning the] appeal are very weak, if not negligible," Judge De Ridder said.
Mr Taylor said the family was saddened by the ruling and would look at options to take the matter further.
He said it was a shame that a family was being kicked out of their home of 25 years.
The family hoped it could persuade HNZ to talk further to find a solution during the time it has left at the property: "We've got 21 days to get HNZ, or some higher authority, to sit down with us and iron out this problem for the sake of that whanau, the Otangarei community and HNZ."
HNZ said it would release a statement on the matter today.