A chainsaw should never start smoking, which can imply a serious, even fatal engine problem. Shut the engine off immediately to prevent exacerbating the problem. Most likely, the chainsaw isn't receiving proper lubrication on the chain or in the engine, or the cooling system and air filtration system are blocked.
Bar Lubricating Problems
If the oiling system on the chainsaw isn't working properly the bar and chain will start heating up from the excess friction. Bar and chain oil prevent the chain from getting too hot, and if the bar's tip starts smoking, the reason is improper chain lubrication. Check the bar's oiling system and clean it thoroughly. Clean out the guide slots on both sides of the bar with a knife or other sharp object. Blow out the bar oiler holes with compressed air and make sure the chain is receiving adequate lubrication.
If the engine is smoking as a result of internal heat, the smoke will generally come out of the muffler. A dirty muffler will have an excess build-up of carbon on the walls, spark arrester screen and around the exhaust port. These deposits of carbon can ignite inside the muffler and re-burn causing the engine to start smoking. Clean the muffler and spark arrester screen with a heavy wire brush and mild detergent. Replace the screen if it's too dirty.
Dirty Air Filter
On the other side of the air system, the air filter helps clean out the air before it mixes the fuel. If the filter is too dirty to clean the air, too little air may be present in the fuel mixture, so when it ignites in the cylinder it ignites too hot. This excess heat can build up and cause the engine to overheat and start smoking. Wash the air filter in soapy water and rinse under cool water. Allow it to dry thoroughly overnight before reusing. Change the air filter regularly, at least once a season.
Dirty Cooling System
A chainsaw uses two places for the air-cooling system. The first is on the starter side of the engine. The flywheel, when it spins pushes cooler air down into the crankshaft to keep it cool. The cylinder, on top of the engine, sucks cooler air in to keep the piston cool while it moves inside the cylinder. Clean both of these areas with a soft brush and gently brush off any dust or build-up. Tilt the engine so that all dust falls away from the inside when brushing these areas off.