A vacuum cleaner must deal with dirt, grime, pet hair and a host of other problematic materials. Although a motor that has gone kaput can give off a burning smell, most of the time the cause of the smell is an accumulation of debris. You do not need any special tools to troubleshoot a simple cause of the smell. Always unplug the vacuum cleaner, though, before you do anything else.
Check the brush roller to see if it's too full of debris to spin properly. Turn the vacuum cleaner over so that you can see the brush assembly. Push an end of the brush assembly back and forth with one hand while you run your other hand along the length of the assembly, removing debris. Reverse the position of your hands, only this time pushing the other end of the brush assembly back and forth as you remove the debris.
Check the connection socket between the vacuum bag and the intake hose. Dirt can seal the socket. Go outside so that dirt doesn't wind up on your floor. Open the vacuum cleaner's bag compartment and remove the bag. Run your finger around the rim of the connection socket to remove loose dirt. Put your finger inside the socket, run it around the inside of the rim, and remove any debris there. Blow out the connection socket with a burst of compressed air. Spray lubricating spray on the rubber ring attached to the socket. Place a new vacuum bag on the socket. Close the vacuum bag compartment.
Let the motor of the vacuum cleaner cool down. Place the vacuum cleaner on its side so that you can access the panel where the vacuum belt is located. Open the panel. Remove the vacuum belt, which may have stretched out, causing the brush assembly to move too slowly. Replace the vacuum belt, and close the panel.