This question comes up a bit in the republic debate, mainly because some people say all sorts of crazy things about the cost of a New Zealand head of state, and make outright false claims about the monarchy (i.e. “It costs us nothing!”).
The truth is that at the moment, our Governor-General costs New Zealand taxpayers about $10m a year. According to the 2018 Budget Estimates, (for Vote Prime Minister and Cabinet, since the Governor-General’s office comes under the Prime Minister’s department):
$4 million on for support services to the Governor-General and maintenance of the official residences;
$2 million for salary, allowances, programme and travel expenses outside New Zealand of the Governor-General;
$3 million for Government House and “Crown Assets”;
$800k for “Government House Capital Investment”
There’s also the cost of the Royal Honours system, but that’s not attributed to the Governor-General’s office. There’s possibly other costs hidden away in other estimates, but for the sake of simplicity it’s about $10m a year. This hasn’t changed much in the past few years, it seems to over around the $10 - $11m mark.
Contrast this with the Republic of Ireland, a country with a president equivilent to our Governor-General in terms of powers, and of similar size population wise to New Zealand. In their 2018 Budget, the President of Ireland cost 4.5m Euros, or roughly $7.7m in New Zealand Dollars.
It’s clear that the cost of our Governor-General is about that of an actual head of state.