Councillor Willow-Jean Prime said Mr Tau was in a position of leadership and had to lead by example.
Taking kukupa went against the law as well as a rahui imposed almost 20 years ago.
"It's not a good example. This is dishonest, and it's bad judgment."
Mrs Prime said some had argued that taking kukupa was allowed by tikanga (custom) but a meeting at Kaitoki Marae, in Motatau, had imposed a rahui on the birds across the Ngati Hine rohe (tribal area) in 1996 because their numbers were so low. That rahui was still in place.
She could not speak for Ngai Tahu, from whose area the birds are thought to have come, but even if Maori had a customary right to eat kukupa that had to be weighed up against kaitiakitanga (guardianship) and the responsibility to protect a threatened species.
Mrs Prime, who is of Te Kapotai and Ngati Hine descent, said the call for Mr Tau's resignation was not related to Ngati Hine's opposition to Ngapuhi's current Treaty settlement process.
However, the kukupa incident did call into question his leadership ability, she said.
Mr Tau has said he will not comment further while the incident is being investigated.
Meanwhile, Mr Harris, who is also a cousin of Mr Tau, said he found it untenable to remain on Tuhoronuku in light of its leader's actions.
He said Mr Tau's "mistake" made a mockery of years Mr Harris had spent educating Northland's young people about the protected native birds.
"To have this happen is an absolute slap in the face," he said. "Sonny Tau needs to face the people, Sonny Tau needs to face Ngapuhi and man up to what he's done."
Tuhoronuku deputy chair Sam Nepia said Mr Harris's resignation "was unfortunate".
