The group said councils in Stockport and Salford had more than 50 complaints but they resulted in just one inspection.
Liz Godfrey, of the campaign group, said: “It just shows the government is not giving the council the powers to do anything about this growing problem.”
Bury Council’s leader Eamonn O’Brien, who is GMCA clean air lead, said: “Before we get to the point of issuing fines there are many, many steps before then to encourage people and notify them that they might be in breach of the local rules.
“We send letters to people but to go down a route of fining people and thinking that’s a good sign, I actually think it would be a bad sign, a sign of failure.”
The councillor said people should be aware that wood burning “carries some risks to yourself and your family and also the wider community around you”.
