Could a quadcopter fly if its propellers were encased in cylindrical tubes? Picture a standard quadcopter, with regular old propeller guards, which are generally just circles around the propellers. Now, take the propeller guards and stretch them so that they're cylinders surrounding the immediate cross-section of propeller blades, as well as some distance above/beneath them.

Now I am fairly certain that it would be able to go up/down. But would it be able to turn? I know it is a relatively vague question, but after thinking about it for some time, I really have no idea. I might just modify the propeller guards on my cheap little drone to see if this works.
Yes. It would be able to turn. Of your 4 propellers, 2 spin clockwise and 2 spin counterclockwise.  The torques cancel out.  To make it spin, you run the 2 clockwise propellers a little faster and the other 2 a little slower.  Now the torques do not cancel out and the quadcopter spins.  This does not change if you have a ducted fan versus a propeller