Sorry, I missed seeing this before.
Yes, as with any change in your circumstances, contact INZ mentioning all your details and any reference number you may have been given by that time. In this case, tell them the new passport, number (whatever), is now in use instead of old passport, number (whatever).
Two things: when you gain Residence, you will have two years' travel conditions (the strange name for permission to enter NZ freely). As the two years come to an end, you need to apply for Permanent Residence, if eligible, or a further Residence visa - see here.
https://www.immigration.govt.nz/new-...anent-resident Otherwise, you will no longer have the right to re-enter NZ from outside. However, while you are inside NZ and not travelling overseas, your Resident visa lasts forever. If someone is outside the country with no travel conditions, in effect, their visa lapses, because they can no longer get back in to take advantage of their right to live and work in NZ. As it says on that page, for a short while, it is still possible to apply for PR or residence to put the situation right, but it is much more simple to think ahead and be onshore when the changeover needs to happen.
Whenever your passport expires, wherever you are in the world, you can't pass an international border. So if you were outside NZ, wanting to travel back, even if your travel conditions were still valid, you would not be allowed to do it. The visa does not depend on the passport - it's something that has been granted to you, the person, and is on record in the INZ computer.
But anyone needs a valid visa AND a valid passport to enter the country, hence needing to tell INZ your new passport number when it has been renewed, and to get whatever visa you are then on transferred to relate to the new passport, which is your identity document.
https://www.immigration.govt.nz/new-...a-new-passport