Air, fuel and spark is what your saw has to have to run.
After you get all the water out of the crankcase and the air filter is dried or removed so air can pass and assuming that your sealed up electronics work when damp, your chainsaw will run after being submerged.
Back in the dark ages of off-road motorcycle racing, before proper shrouding of the intake and sealed electrical systems, it was not uncommon to be seen on the side of the trail after a water crossing with the bike turned upside down on its bars, sparkplug removed, pumping water out of the crankcase. After that, if it was a points and condenser system, you took the side cover off and hoped the engine heat that remained after the dunking would dry them while you wiped with a rag to help things along. Fifteen or twenty minutes later you'd be up and running again. Sometimes quicker.
Pretty soon though you figured out how to shroud the intake so this didn't happen so often. Which worked fine until you hit a slick rock and fell over in the creek and it sucked water while on its side. Or you picked the side of the trail with the 3 foot deep hole in the creek and went under. Then you got to push up on the bank and execute the procedure detailed above.