Phillips suffered the injury while playing for Washington Freedom in the Major League Cricket (MLC) final on July 13.
When he was assessed upon arrival in Zimbabwe, it was determined that he would require "a number of weeks" to recover, according to an NZC statement. Tim Robinson, who was a late addition to the T20I squad for the tri-series as cover for players involved in the MLC knockouts, will remain with the side for the rest of the tri-series.
Phillips' replacement in the Test squad will be named in due course.
"It's obviously disappointing to lose someone of Glenn's calibre," Rob Walter, New Zealand's head coach, said. "Much like Finn (Allen), we really feel for Glenn and that he misses out on this series.
"We know he was eager to get out on the field for the Blackcaps and unfortunately, he won't be able to do that for this series. We know he'll work hard to get himself back onto the park, and I look forward to when that happens."
Phillips will return to New Zealand along with fellow allrounder Jimmy Neesham and wicketkeeper-batter Mitch Hay who were also called in as cover for the start of the tri-series alongside Robinson.
Phillips joins Ben Sears (side injury) and Finn Allen (foot injury) on the sidelines. New Zealand, though, will be boosted by the return of left-arm fingerspinner Ajaz Patel for the Test series. Having worked his way back from a serious knee injury, Ajaz proved his form and fitness for Central Districts in the Global Super League (GSL) in Guyana.
New Zealand have started their T20I tri-series campaign with a 21-run win against South Africa in Harare.
Senuran Muthusamy wants to embrace the challenge of putting in consistent performances at the highest level as he chases a place in the Proteas squad for next year's T20 World Cup.
New Zealand made it two wins from two in the T20I tri-series in Harare after dispatching host nation Zimbabwe by eight wickets on Friday.
The Junior Springboks are set to face New Zealand in the final of the World Rugby U20 Championship on Saturday evening, in what promises to be a thrilling encounter.
The All Blacks wing stocks have been dealt another blow on the eve of the third Test match against France at the FMG Stadium Waikato in Hamilton.
