Burns captained the side who booked a maiden T20 World Cup berth in July through European qualifiers, setting up the nation's tournament debut in India and Sri Lanka from February 7.
However, the federation said Burns will not be a part of the World Cup squad, with South African-born Wayne Madsen to lead the side instead.
"Although preliminary contacts regarding his (Burns) availability have been made in recent weeks, no final agreement has been reached and, as a result, no contract has been formalised," the FCRI said in a statement.
"At the same time, internal technical and organisational assessments have led to Wayne Madsen being identified as the most suitable choice to lead the team."
The federation did not elaborate on what issues concerning team stability had influenced their decision but said Madsen will lead Italy in the upcoming tour against Ireland before they head to the World Cup.
Burns played for Australia between 2014-2020, making his Test debut against India on Boxing Day 2014. He accumulated 1,442 runs in 23 Test appearances.
He also turned out for the Melbourne Stars in the Big Bash League and Queensland. He announced he was moving to Italy in May 2024, as a tribute to his late brother Dominic, who had died three months earlier.
In an emotional post on Instagram, Burns, who honoured his brother by wearing his playing number, 85, on his shirt, said: "The days, weeks and months since my brother's death have been the toughest I could ever imagine.
"I'm not too proud to admit that it's been a daily battle that I often lose.
"While a part of my soul feels like it will always be missing, I know this shirt will carry on his spirit and give me strength."
