The airline had scheduled additional services later in the week.
"The airline apologises for the disruption and will continue to monitor the situation closely," it said.
"The volcanic conditions in Tonga are not currently affecting Air New Zealand services to other destinations."
Andrew Allison, who is in Tonga on holiday, said his flight had been cancelled twice due to the ash cloud.
The ash cloud as seen from the main island. Photo / Andrew Allison
The Hunga Tonga-Hunga Ha'apai underwater volcano - about 30km southeast of Falcon Island volcano - has been having small eruptions over the past few weeks, but a larger eruption in the past few days has resulted in a huge plume of ash covering the area.
An advisory has been put out by the Volcanic Ash Advisory Centre in Wellington.
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New Zealand's Civil Aviation Authority is keeping a close eye on the volcanic activity.
Meteorological manager Peter Lechner said: "Looking at the advisory, they've still got an ash cloud extending from the southeast.
"From the ocean surface, the plume is up to 30,000 feet (9km) high."
The volcano is about 63km north of the island kingdom's main island and capital city: Nuku'alofa.
The advisory says it is expected the ash cloud will dissipate by about midnight (NZT) tonight.
The Wellington Volcanic Ash Advisory Centre shows the reach of the plume. Photo / VAAC Wellington