The emissions that change climate include various gases and the colors of smoke. If the wood used for cooking is 100% renewably harvested, the emissions of CO2 can be carbon neutral. Why? CO2 is absorbed when the plant grows and the same amount of CO2 can be released when biomass is burned. 

100% renewability can help CO2 to become climate neutral. However, the fraction of non-renewability (fNRB) does not change the amounts of other climate forcing emissions. Smoke is smoke. Etc.

What are the most powerful cook stove emissions affecting climate?

In general, adding methane and carbon monoxide to Carbon dioxide (CO2) adds a bit to the total warming influences (CO2e). 

However, adding short-lived climate forcers such as NOx, SOx and Black Carbon to the above has been estimated to more than double the warming potential. 

For this reason, it seems to be important to add the short-lived climate forcers when calculating how to address climate change with cook stoves.