What’s the law regarding junk mail?

There’s no New Zealand wide law for distribution of printed mail however some regions have bylaws covering this. The included laws below are not complete, I am not a lawyer and the information provided does not constitute legal advice.

Auckland City Council Bylaws

28 Unaddressed mail
(1) No person may deposit, cause, permit or authorise the deposit of any unaddressed mail, advertising material, community newspapers, clothing donations bags, circulars, leaflets, brochures, samples or flyers in any letterbox which is clearly marked “addressed mail only”.
(2) Subclause (1) does not apply to:
(a) public notices from government bodies, local authorities or New Zealand Post;
(b) election material during the period beginning two months before polling day and ending with the close of the day before polling day.
(3) No person may deposit, cause, permit or authorise the deposit of any advertising material, clothing donations bags, circulars, leaflets, brochures, samples or flyers:
(a) in any letterbox which is clearly marked “no circulars”, “no junk mail”, or words with similar effect;
(b) on a vehicle parked in a public place;
(c) in a letterbox in unsecured circumstances, if it is likely to escape and become litter.
(4) Subclause (3) does not apply to:
(a) any daily or regular newspaper, community newspaper or magazine;
(b) public notices from government bodies, local authorities or New Zealand Post;
(c) communications from local community organisations, charities or charitable institutions;
(d) election material during the period beginning two months before polling day and ending with the close of the day before polling day.
This is from the Auckland City Council Bylaws Part 22 – Solid Waste (2012).

We have the Unsolicited Electronic Messages Act which covers email, instant messaging, TXTs & PXTs of a commercial nature. It does not cover faxes, Internet pop-ups or voice telemarketing. The Department of Internal Affairs enforces this and has an Anti Spam section on their website.