You Do Need power plug adapter while traveling from Burundi to New Zealand.

Here is Why You Need An Adapter

The outlet and plug type used in New Zealand is I and in Burundi are C and E. This is how they look like.

Burundi

In Burundi, outlets of type C and E are in use.

Type C

Type E

New Zealand

In New Zealand, outlets of type I are in use.

Type I