Burney was a spectator at the game, and reportedly took exception to a heavy challenge against his teenage son.
Both Forrest Hill-Milford and North Shore United were fined $200.
Auckland Football Federation chairman Bob Patterson said the incidents were both extremely serious.
"We fully support the decision of the Disciplinary Chair in determining the sanctions to be applied in both instances."
"We view the NRFL as the premier winter football competition in the country and incidents such as these, in a competition which serves as a key step on the international pathway and helps to shape the behaviour of our aspiring young footballers, will simply not be tolerated."
"Ahead of the 2019 season, AFF and the NRFL management will continue to work with all clubs and stakeholders to ensure everyone involved in our game is aware of their role in delivering a competition of the highest standards on and off the field of play."