For most two-strokes with that difficulty I might suspect a plugged carburetor. The problem with many home tools is that they get infrequent use and the fuel (gas) in the carb can evaporate, leaving behind a residue of what is often referred to as varnish. It's not, but it appears as thick as paint and is a dark brown. It plugs the tiny openings in the carburetor jets and even some interior passages.
If it was perhaps three or more months since you last operated your chainsaw, I would certainly diagnose it as a plugged carb.
To fix, remove the float bowl and using a auto part store carb cleaner, vigorously spray into the openings in the bottom of the carb. Additionally, back out any screws you see that would be adjustments (carefully counting the turns until it is free) and spray into those openings as well. Let the carb sit for perhaps 20 minutes, and repeat the spraying. Do not be frugal with the spray. You need lots of it, close, and at full pressure.
Be careful with the spray as it can be damaging to you, plastic parts, and will cause rubber to swell enormously and become brittle. Do not let it remain in contact with those parts for long. Additionally, protect your eyes and avoid inhaling the product.
After the 2nd spraying (third if the motor has not been started in six months or more) clean out any remaining carb cleaner with a rag, reassemble the carb, put NEW fuel in the tank and try your engine again. If done correctly, this may correct the problem. If not, you may have to disassemble the carb. I was avoiding that as many inexperienced people can create more damage than good working on small carburetors.
Under any circumstances, DO NOT attempt to run a wire or drill through the delicate sized jet apertures (holes).
That usually does it.
Cojon, December 2008