It's a very tough question to answer because it's very hard to predict. I have some model of the general effects of allowing greater immigration, but the predictive value of the model breaks down the more radical the change.

- I see letting in anyone with the equivalent of a bachelor's degree (or say anyone that anyone is willing to hire for more $40k a year) as basically having no downsides at all. Empirical data on immigration reveals that it has no impact on wages except at the very low end, among unskilled high-school dropouts.

- As you allow lower incomes, welfare tends to become an issue. Currently, people do not immigrate in the US for the welfare (illegal immigrants tend to be net taxpayers, even when including healthcare) but people do immigrate to Europe for the welfare, so it's something that happens. However, if you take a country like France (where I grew up in), welfare payments aren't the biggest drag on the economy. Taxation is high, but other places with similar levels of taxation fare much better. Regulatory drag, in particular in the job market is a much bigger concern. In turns, this creates unemployment which aggravates any welfare problem. Evidence shows that as societies become more culturally and ethnically mixed, they tend to be less and less in favor of welfare. So higher immigration of welfare seeking immigrants may undermine popular support for welfare. It's a whole other discussion, but I'd consider disappearance of state funded welfare as a good thing. In addition, it's also possible to selectively deny welfare to immigrants, or to require a twenty year residency period before becoming eligible.

- Political externalities are often mentioned. Won't immigrants vote for the same policies that made their home country a living hell? I find this doubtful for two reasons:

  a) The evidence points that in most democracies, the policies that end up being enacted are the ones favored by a wealthy elite, not the ones that have popular support. If you compare the actual policies of the US compared to the views of the median voter, they are strikingly better than you'd expect. How this comes to be is unclear, but the effect is important.

 b) The favorite political view of most people is the status quo, or some very minor version around it

Now getting to New Zealand. Assuming only honest hard working immigrant seeking a better life? You've cut out the work for me! I think you'd see a lot of high rises build up in New Zealand. Consider that the current population of New Zealand could fit in a circle 7 km in radius in a city with the density of New York. If Stewart Island was all built up, it could hold the entire population of Canada.

In practice, as more people would emigrate, real estate prices would appreciate, to the point where it wouldn't be economical for immigrants to come in, even if the borders were open.