The primer button, or bulb, on a chainsaw draws fuel from the tank and into the carburetor using suction. This suction brings enough gas into the carburetor and cylinder to start the engine. Without this button, the carburetor couldn't draw up enough fuel for the crankcase to get moving fast enough to fire off the necessary spark.
Bad Gas Lines
When the gas leaves the fuel tank it enters a plastic hose that connects the tank with the carburetor. Another fuel hose brings any unused fuel back into the tank. The primer bulb sits in the middle of this return line on most chainsaws. Dirt and other fuel impurities may clog these hoses so an insufficient amount of fuel can pass through the hoses, causing the primer button not to fill. Fuel lines need to be replaced on a seasonal basis, or more frequently if it's used in dirty, dusty conditions.
Faulty Primer Bulb
Another cause of the primer's filling problem could be the bulb itself. Small perforations on the rubber could inject enough air into the fuel system to make the suction too weak. Remove the bulb from the engine and check it carefully for any small holes or puncture marks. Replace any worn or cracked bulbs. Check the points on the underside of the bulb as well. These two holes can also get clogged or be blocked by dirt in the fuel.
The fuel hoses connect to two elbow connector joints on the carburetor. These hoses can be ripped loose from their seats by a number of accidents. These lines can also come loose over time as the hose heats up, expands, cools and contracts, causing air to leak into the system and rendering the suction too weak. Check the fuel hoses' connections with the carburetor. Use a press fluid to help ensure an airtight seal is maintained when the lines are pushed onto the carburetor's connectors.
Problems in Carburetor
The carburetor in most chainsaws will use a set of diaphragms to help pull fuel into the carburetor, mix and measure it and send it off to the cylinder. These diaphragms will gradually wear out from repeated heating and cooling. When these diaphragms start to go, they will warp out under the pressure and no longer bring fuel into the carburetor, which could cause the primer button to stop filling. Carburetor diaphragms should be replaced by a professional.