Chainsaws will indeed work in the sub-arctic. My chainsaw, a Stihl 046, has the arctic option, which uses the magneto to provide electric heat to the handles and carburetor. The coldest temperature I've used it at was -35C, and it ran just fine. Frozen trees are harder to cut, and heavy winter clothing is cumbersome, so everything takes longer, but if the chainsaw is well tuned and filled with clean fuel - because even a little moisture is gonna cause issues - it will do the job.
The limit might come at around -45C, when steel begins to become brittle from the cold. But that's too cold to stay outside for long anyway, so the limit of human endurance will probably come before the machine's.
Of course, this only applies to well maintained professional grade chainsaws like Stihl and Husqvarna. Don't expect the el-cheapo brand that you got on sale at the local big box store to start at -20. It wasn't made for that.