It is always worth looking at the website for the relevant district or city council (see
maps and list of
council websites) to see if they have a cemetery database online - many do. Here are some I am aware of and have used successfully (organised from north to south):
Napier (some photos)
Palmerston North and surrounding rural areas
Wellington City and
Hutt City (different parts of Greater Wellington)
Waimakariri (the area immediately north of Christchurch)
Christchurch City (most of the entries for the Barbadoes Street cemetery, one of the older burial areas, are missing)
Ashburton and some other rural Mid Canterbury cemeteries (this is in the form of alphabetical listings by surname)
Timaru (with photos of many of the gravestones)
Waimate, South Canterbury (see database link at the bottom of the page)
These sites usually do not include any listings from church graveyards, but rather are listings from the municipal cemeteries - but most NZ burials will be in those municipal cemeteries. There is a more complete listing of NZ cemetery sites
here, which looks to have been updated recently, but I haven't tested its links.