Mr Pora was convicted of her rape and murder in 1994 and again in 2000.
Mr Pora was released on parole after spending 21 years in jail.
Malcolm Rewa was convicted of her rape in 1998 after DNA from semen was linked to him. But two juries could not reach a decision on whether he murdered her.
Rewa is serving life for the rape of Ms Burdett and attacks on 24 other women.
Mr Pora's lawyer, Jonathan Krebs, said they were pleased by the decision. He had earlier said compensation could be looked if there was no retrial.
The Crown did not seek a retrial for Mr Pora. It said in submissions that it was not in the public interest even though there was sufficient evidence for a retrial.
It said Mr Pora had already spent 20 years in prison, a period consistent with terms served for murder.
While new evidence that Pora suffers from foetal alcohol syndrome disorder provided a "theoretical basis" that Pora's confessions were unreliable, it would be open to a third jury to conclude he was a party to the crime "having regard to the totality of the evidence", including an expert evidence brought by the Crown.
Ms Burdett's brother, Jim Burdett, said he was pleased Mr Pora would not be retried. He said he had come to the conclusion some time ago that he was not involved.
He had an open mind about whether Rewa should be tried a third time for his sister's murder but did want justice for her.
"With the quashing of the convictions, in principle it is the same situation [as in 1992], it means the murderer is still out there."
Fight for freedom
1992, March 23: Susan Burdett raped and murdered in her home in South Auckland.
1993, March 23: Teina Pora charged with burglary, sexual violation and murder.
1994, June: Pora convicted as a party to the rape and murder on the basis of confessions he made. Sentenced to life in prison.
1996, May: Rewa arrested after attacking a young woman in the inner Auckland suburb of Remuera, DNA from Rewa's father found to match semen from Burdett crime scene.
1998: Rewa eventually convicted of the rape of 27 women, including Ms Burdett but two juries fail to reach a verdict on murder.
1998, May 30: In 1998 Rewa was convicted on multiple sex charges dating back to 1987 and sentenced to preventive detention with a minimum non-parole period of 22 years. He was convicted of the rape of Ms Burdett the following year.
1999: Court of Appeal quashed Pora's convictions as a result of the DNA evidence implicating Rewa and evidence that Rewa acted alone.
2000, June: Pora was again convicted at his retrial, based on his confessions and witnesses, some of whom it later emerged were paid. His appeal to the Court of Appeal was dismissed.
2009, September: Private investigator and former police detective Tim McKinnel visits Pora in prison and is given permission to make inquiries on Pora's behalf.
2011, September: Pora team file notice of application for the Royal Prerogative of Mercy but two years later are granted an appeal to the Privy Council.
2012, May: Police's criminal profiling expert goes public in Herald with view Pora not involved; Pora's team sue police claiming it is unlawfully withholding evidence, Ms Burdett's brother says Pora is innocent.
2013, February: It is revealed police paid some prosecution witnesses.
2013, August: The Police Association calls for an independent inquiry into Pora's convictions.
2014, April: Pora granted parole at his 13th appearance before the board and after spending 21 years in jail.
2014, November: Five-member Privy Council panel hears appeal.
2015, March 3: Privy Council quashes convictions.
