How dangerous can it be if something falls from a flying aircraft?
The standard operating procedures state that before every flight, a thorough pre-flight inspection should be performed. Depending on the aircraft, minute and tiny details are not even omitted.

Things do not normally and occasionally fall off from a flying airplane, the same way as different objects do not fall from a driving car, although the pre-drive inspection of a car is no where nearly as comprehensive as the pre-flight inspection. Yet, how many times we have seen a bumper, guard, muffler or rear-view mirrors falling off?

How dangerous it would be when it happens? This depends on the part which has fallen. Some parts are not noticeable until landing or next inspection and some can cause disasters.

Baltimore Sun reports this:


  No deaths or injuries have been reported in such incidents, but the
  Federal Aviation Administration is not even 100 percent certain of
  that.


The proper term for this incident is Things Falling Off Aircraft (TFOA). The pilot must then submit a hazardous material report within 24 hours in case the material can cause further harm.

This site has listed some TFOA incidents:


  
  In August 2000, a KLM Royal Dutch Airlines Boeing 747 was forced to make an emergency landing when some engine pieces plummeted to the
  ground and landed on a crowded Los Angeles beach
  On March 10, 2005, a couple taking a stroll through woodlands near Gatwick airport had a lucky escape when a door from a passing British
  Airways jet, crashed to earth. A part of the 70-pound door from a
  Boeing 777 had become dislodged just after take off and missed the
  couple by only 20 feet! 
  On August 11, 2006, a door fell off from a TAM Fokker 100 from Sao Paulo to Rio de Janeiro. It dropped off 18 minutes after departure but
  the plane was able to return to Sao Paulo without incident. Nobody was
  injured.
  An engine exploded on a Japan Airlines flight bound for Tokyo with pieces of the engine falling on a neighborhood in Jakarta. The falling
  debris hurt no one but 16 homes were damaged.
  The right wheel of a Blue Panorama Airlines Boeing 737 fell off after it departed from an airport in Spain and was forced to make an
  unscheduled landing amid emergency in Rome. 
  A similar incident happened near Los Angeles where an American Airlines Boeing 757 lost its right nose gear but managed to land
  safely with 105 passengers on board. 
  A landing gear door fell off a Delta Airlines Boeing 727 shortly after the plane took off from Boston*s Logan International Airport.
  The 30-pound metal door crashed into an empty street in a quiet
  residential neighborhood.