Creating excessive smoke pollution is an offence under the Protection of Environment Operations (POEO) Act 1997 (s135). A plume of smoke from a chimney should not be visible for more than 10 minutes and should not extend more than 10 meters.
Talk to your neighbour
The owner may not be aware their wood heater or barbecue is smoking excessively, so discuss it with them first.
Tip: All slow-combustion wood heaters tend to smoke at light-up or refuelling. They should not smoke excessively for long periods if they are operated correctly and are well maintained.
Contact your local council
Issues with excessive smoke from domestic wood burning can be reported to the local council. They can issue smoke abatement notices for residents who do not make an effort to prevent excessive emissions of wood smoke.