No, this will unfortunately (?) not work as advertised.
The basic problem is lift. There are three basic ways I can see a chainsaw mechanism generating lift, and all have insurmountable issues.
One is basically using the mechanism of the saw itself to push air down with, effectively, little scoops instead of teeth. This might work in water, which is a lot heavier (paddle steamers use this to travel forward rather than upward), but with air the volume you'd need to push down would far exceed your available power.
The second, somewhat more practical, is to turn each individual tooth of the chainsaw into a wing, which generates lift as it turns. This is basically how helicopters work, except that their blades only go around in a circle instead of in a pill shape. The problem here is that even if you could get it to work (and the connection of the teeth to the "hub" seems very suspect) it will look and act much more like a helicopter blade than a chainsaw: the "teeth" will be long and thin, and rotating extremely quickly. Trying to saw through things will put immense torque on the teeth and probably break them.
The third option is to taper the profile of the teeth and saw such that the saw itself is a wing. This approach will generate lift, but has issues of its own. One is that the edge of the wing, where the teeth are, is normally where the control surfaces are. Having a bunch of fixed, moving slats in your wing is going to lead to some profoundly odd handling characteristics. It will not fly even remotely like a normal plane.
More problematic to the wing approach is that chainsaws are not designed to saw through things at full speed, the way you see in zombie movies. They're designed to be pushed with slow, constant pressure like any circular saw. (Think of a table saw: you don't throw boards at it, you push them gradually into it.) Any practical design will have a stall speed that's far faster than its safe chainsawing speed; if it's going fast enough to generate lift, it won't be able to cut through objects fast enough and will become stuck or will fling objects away at high speed.
A vehicle that both flies and chainsaws may be possible (though probably not practical) but a single mechanism that both generates lift and is a usable saw at the same time is not.