Are you sure the fuel tank cap is supposed to be sealed? I don't currently have a chainsaw to look at, but my line-trimmer has the same requirement as a chainsaw to run with the engine in any position, and it has a leakproof fuel cap that is not actually sealed. If you look at the underside of the fuel tank cap you may find a sort of plastic dome with a small hole in the center. If you remove the plastic dome you will find a cunning device, similar to the device on a car air conditioner's Evaporator-Blower Assembly for letting water out. (It is known in automotive circles as a 'bloop tube' for obvious reasons: when enough water accumulates it suddenly opens, drops the water out onto the ground under the car, and re-seals itself). The one on the line-trimmer fuel cap lets air into the tank: when enough vacuum is generated it suddenly burps and lets in a short burst of air, then seals again. If the device is allowed to dry out for a long period it is prone to sticking closed: it ends up with a film of oil in it, and the oil oxidises, leaving a sticky film.
The blooper is quite tiny: about 3 mm diameter, with a squeezed-flat section. If it is working, you can suck air through it but can't blow through it. If you can't suck or blow through it, try squeezing it from the two sides until it pops open, to unstick it. Washing it in petrol would help to remove the sticky oil film.
