Have you cleaned/replaced the air filter?
Mine can get caked solid when cutting wet wood.

I believe that saw has a chromed lined cylinder. If the lining starts to peel off or is scored the engine will run when cold but once it is warmed up it will not run until it has cooled back down. It does not take much heat for this to happen. Take the air cleaner off and look into the carb with a flashlight. You should be able to see into the cylinder. You may have to move the piston to the bottom of the cylinder but you should be able to see enough of the cylinder to determine its condition. If it is a lined cylinder and is damaged sorry to say it may be cheaper to replace the saw.