Team NZ said its management and Burling had been in discussions since the third successive Cup triumph against Britain in Barcelona last year but were unable to reach agreement.
"Pete has been a central figure who has grown incredibly within Emirates Team New Zealand since he joined 10 years ago," Team NZ chief executive Grant Dalton said in a statement today.
"We can all look back with great pride on what has been achieved, having enjoyed unprecedented success as a team with Pete at the helm."
The decision comes days after INEOS Britannia withdrew its intention to challenge for the next Cup, having split with skipper Ben Ainslie in January.
Briton Ainslie has vowed to compete in the 38th edition with his British America's Cup team now known as Athena Racing.
Burling joined Team NZ after winning gold and silver medals at the 2012 and 2016 Olympics sailing with teammate Blair Tuke.
He helmed New Zealand to victory over defender Team USA in Bermuda in 2017, becoming the youngest skipper to win an America's Cup at the age of 26.
Four years later, he guided New Zealand's successful defence against Luna Rossa in Auckland while juggling preparations for the Tokyo Olympics where he won another silver medal in the 49er class with Tuke.
Team NZ did not specify why discussions had broken down but Dalton suggested there may have been a sticking point over Burling's role going forward.
"What has enabled winning in the past does not always equate to winning in the future, especially in much tighter America’s Cup cycles which require a dedicated and new approach for continued success," he said.
Burling said in the Team NZ statement he would "take a moment to reflect" without elaborating on his plans.
"I’m immensely proud of everything we’ve achieved together and grateful to everyone who has been part of this chapter," said Burling.
(Reporting by Ian Ransom in Melbourne; Editing by Shri Navaratnam.)
