The relationship between a forest fire and local weather is very complex and almost symbiotic in nature. The fire and the weather continuously play off of one another. Often the weather created by the fire contributes to the growth of the fire. This is how a small forest fire can quickly grow out of control and into a wildfire. Fire creates a number of variables that can play a role in weather changes. Size and intensity of the fire determine how much it will affect the weather. The weather conditions leading up to the fire can be exacerbated by the fire itself. The terrain of the location of the fire also plays a role in what chemicals end up in the air during the fire.
In the most simple example of a forest fire changing local weather, a fire is hot. The high temperatures from the fire heat the surrounding air. That hotter air rises into the atmosphere. Other air on the surface rushes in to replace the rising hot air in a process known as convection. This convection can create localized winds that can easily become intense enough to spread the fire even more. In this sense, a wildfire will create its own weather that is conducive to its life and growth.
Another simplistic example involves the interplay of water vapor. The combustion involved a forest fire produces water vapor as a byproduct. The water vapor gets trapped in the convection currents and becomes cooled slightly as it rises. When water vapor cools it condenses. As the vapor condenses and cools it will release latent heat and reinforce the convection already taking place. The process is similar to the one that takes place in a convection thunderstorm, and like in a convection thunderstorm lighting and downbursts can be created without warning. Both the downbursts and heat lightning can, in turn, contribute to the spreading and growth of the fire.
Wider impact of fire
Sometimes during the convection when the hot air is rising, smoke and particles from the fire are also taken into the air. Depending on what the fire has burned, other things like chemicals can also be lifted. Those chemicals in the atmosphere are comparable to sulfur being cast into the air by volcanic eruptions. A large forest fire can send chemicals much higher into the atmosphere than a volcano, into the stratosphere. Once in the stratosphere, those chemicals will spread around the world affecting world climates not just the local microclimate. A forest fire can thus have influence much larger than just on local weather. They can impact world weather as well.