My Europe-based company has offered to allow me to work during my holidays in New Zealand which would really help out.
I go there the 1st June.
Do I need a special visa or is a tourist one OK?
I'm living in NZ, what I can see from my experience if your company is Europe-based and you come to NZ with a tourist/visitor visa and you work remotely with your computer for that company then, there wouldn't be any problem.
Now the problem is if you come to NZ and you are trying to work for a NZ company and or doing some freelance job here, then you need a work visa for that and you have to pay taxes.
NZ Immigration website: https://www.immigration.govt.nz/new-zealand-visas