As a kiwi, I can honestly say the MAF control guys at NZ airports are incredibly strict, but rightly so, as we have a lot to protect.
As Nev75 has rightly pointed out, untreated nets, felt soles and any gear that has been in the water needs a good dose of anti-bacterial spray or similar. Just remember to declare your gear as you go through customs, treated or not, as a fine or even confiscation will most likely be the outcome if they catch you out.
Flies wise, fur is really the only thing I would imagine they to scrutinise. Zonker strips, that sort of thing, as it's basically skin and hair which may or may not have been treated. Also, in case you were taking a fly tying kit with you, a hares mask is probably pushing it
You could always leave some of the hairier stuff at home, and buy some great NZ made flies, support the local fly fishing industry and all that
Enjoy your trip!