Children’s services at the authority previously failed repeated Ofsted inspections, and a recent review heard staff and partners describe council culture as “intimidating and toxic”.
The government-appointed commissioner Andy Couldrick found there was blame culture in the Labour-dominated council and a failure to take responsibility.
Conservative councillor Liam Billington told Radio Manchester the Labour group in Tameside were in “such disarray and chaos”, adding he felt they were “no longer fit for purpose”.
But a Labour spokesman said a campaign improvement board was being sent to intervene and oversee immediate changes within the council’s labour group.
Cooney, who has been a Tameside councillor for 34 years, said he accepted that “in order for children’s services to go forward, it can only be done with the support of all Tameside members”.
He said the “events of the last few weeks” made it clear he no longer had that support.
The departures come days after the council’s chief executive Sandra Stewart quit the authority.
In a statement seen by the Local Democracy Reporting Service, Ms Stewart said she stepped down after agreeing with Cooney to become chief executive of Greater Manchester Pension Fund.
