In a reference to Australian Immigration Minister Peter Dutton, who was this month caught joking about low-lying islands being flooded by rising seas, Reverend Naisali said climate change was no laughing matter.
"People may laugh and ask questions, is it real? It is not a matter for us to laugh at, it is real, and we want New Zealand to do more and be more proactive."
Reverend Naisali said he himself had moved to New Zealand to give his children a better future, and that was what Mr Teitiota had done.
His children were born here, had started to receive the benefits of a New Zealand education, and Mrs Teitiota worked as a teacher.
Send the family - and the New Zealand-born and raised children - to Kiribati was akin to putting someone on dialysis on a plane, despite knowing there was no medical help where they were going.
"There is no employment opportunities in Kiribati, there is population density in Kiribati, there are no education opportunities for the children," Reverend Naisali said.
Asked about the case this morning, Mr Key said Mr Teitiota was not a refugee, but an over-stayer.
"There are plenty of people living on Kiribati, and President Anote Tong would confirm the fact that he can live there like anyone else," Mr Key said.
"He chooses not to live there and I can understand those reasons. But to claim he is a refugee I think is just not correct."
Mr Key said there were a number of over-stayers in New Zealand from Kiribati and Tuvalu.
Future New Zealand Governments may need to examine the case for climate change refugees from the Pacific, he said - but the time for that was not now.
"I am certainly not ruling out that a future Prime Minister and a future Government wouldn't take that compassionate view, and I suspect actually that they would. But it would be on genuine grounds that they actually can't live in their country."
Labour MP Phil Twyford, whose electorate includes Mr Teitiota's home in Ranui, said focussing on the issue of whether he was a refugee was disingenuous, as the courts had already decided that was not the case.
Instead, Associate Immigration Minister Craig Foss needed to intervene on humanitarian grounds, particularly because the children were New Zealand-born.
Mr Foss expects to be briefed on the case and hopes to make a decision today.
Green Party co-leader James Shaw said the case was the "canary in the mine", and there would soon be "a flood of people from the Pacific Islands" because of climate change.
"[They] are going to be looking for places to live as their islands sink beneath the sea. And that is the real tragedy and we have no plan for that."
Kiribati's capital, Tarawa, is struggling with rapid population growth.
In South Tarawa about half of Kiribati's 110,000 population live within an area of 13km2, with most living in slum-like conditions.
New Zealand is working with other donors to try and address issues, including by improving access to fresh water, upgrading sanitation facilities, and developing a new subdivision just south of Bairiki Airport.