Long-term exposure to burning wood for heating and cooking can cause heart and lung disease.[1]
The main air pollutants in wood smoke are particulate matter (PM), carbon monoxide, carbon dioxide, nitrogen oxides and a range of other organic compounds like formaldehyde, benzene and polycyclic aromatic hydrocarbons.[1][2]
Using reverse-cycle refrigerant heating (or 'heat pumps') is probably the least worst polluting method of heating. Modern heat pumps can generate (well, move) four times more heat energy than used to drive the pump[3], with the added benefit of moving the source of pollution away from populated areas, or being carbon neutral via window, solar, hydro, geothermal.
Heat pumps only really work well when they have access to fairly warm temperatures. They are kind of worthless in the far north and direct fuel use is significantly more efficient.
If you burn coal for power at 40% efficiency and run a heat pump at 400% efficiency that's only 1.6x heat gain ignoring transportation losses. Drop that heat pump down to 150% efficiency and heating oil can become a better option.
In most areas using solar thermal heaters with backup resistance heating is actually the best overall option. But, in most of Alaska heating oil is much more viable.