Bluebottles. They WILL kill you with their stings and are a plague on NZ beaches.

The Katipo, called Redback in Aussie will hurt you bad but death probably will not happen as the hospitals know how to treat the bite.
Get bitten or scratched even by a Kiwi Possum and chances are, left untreated you will become very sick indeed as they carry all sorts of diseases.
Along with wild Pig, get gored in the bush by one of those and a Katipo bite will seem like good news.
No, snakes are not a problem but some plants are. Like every country NZ has a wide range of Flora and being scratched by some leaves can get you into hospital.
The stats on any of it happening are so low however you shouldn't have a problem.
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addition : New Zealand isn't home to any poisonous animals, apart from the spiders i mentioned. You would most likely encounter white tails and not the other two. NZ is also home to flies, mozzies bugs and other non poisonous spiders etc.