Mr Gunner announced his decision on Thursday, citing personal reasons.
"Some big personal news. I have made the decision to leave Fortescue," he said in a Facebook post.
"Working for Andrew and Fortescue was an incredible opportunity - a dynamic company with a bold green energy vision."
The retired Labor politician, who lives in Darwin, joined Fortescue Future Industries in October 2022 after stepping down as the territory's Labor leader several months earlier.
He said the travel and time away from his wife and children had become too difficult to manage.
"I will always be a friend of the Fortescue family and an advocate for Fortescue and the path they are forging," he said.
"I'm taking a bit of time before deciding with the family what I do next and am now reporting for lunch box duty."
A Fortescue group spokeswoman confirmed Mr Gunner's resignation and said it wished him well.
Mr Gunner served as the NT chief minister from 2016 until he resigned from the post in May 2022, leaving politics soon after.
His departure is the latest in a slew of leadership exits from the Fortescue group, which has also included Fiona Hick, who left after six months as Fortescue Metal's Group chief executive.
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