How dangerous is airborne debris?
The FAA's advisory circular on Airport Foreign Object Debris (FOD) Managament mentions plastic bags and other trash as a type of FOD and says these are the dangers:


  FOD Hazards.   FOD can severely injure airport or  air carrier personnel or damage  equipment.  Types of potential damage include: cutting aircraft tires; being  ingested into engines;  or becoming
  lodged in mechanisms affecting flight  operations.  Personnel injuries
  or even death  can occur when jet blast propels FOD through the
  airport environment at high velocities.


Since you're asking specifically about airborne items, that would mean the most likely dangers from that list are engine ingestion or "becoming lodged in mechanisms affecting flight operations".

The main issue with a plastic bag seems to be that it damages the engine, rather than stopping it entirely. This paper recommends an inspection if any "soft material" (including plastic bags) are ingested:


  
  Remains on impellers and/or LP diffuser exit ducts confirm a bird or soft material has passed through the Engine and possibly
  contaminated the secondary air system, fuel nozzle and HP turbine vane
  cooling air passages.
  If contamination occurs, hot section components deteriorate rapidly, increasing the cost of refurbishment. Therefore, operators are
  recommended to carry out a hot section inspection (HSI) as soon as
  possible after a bird strike or ingestion of soft material is
  confirmed
  


As for affecting flight instruments, the main danger is probably that the bag covers the pitot tube, resulting in false airspeed readings. That can be catastrophic if the pilots don't realize what's going on, but many larger aircraft have multiple pitot tubes so it would probably need a lot of plastic bags (or a lot of bad luck) to cause an accident. And a bag stuck on the windscreen could block a pilot's view but if there are two pilots then that shouldn't be critical.

Of course, all of the above applies to airliners. If you fly a light piston single through airborne debris the results could be worse because of the lack of technical (and perhaps crew) redundancy.