Have we ever observed a body, such as a large asteroid, "hitting" the Sun? Some other SE questions about launching ICBM's into the sun got me wondering whether we have ever observed an object on a path that intersected with the Sun? How close did it get?
Yes

and here's a video of "a Giant Comet Hitting the Sun":

https://www.youtube.com/watch?v=Mat4dWpszoQ

The impact occurred sometime during May 10-11, 2011.  The comet was not named but believed to be a member of the Kreutz family of comets

Many close calls

Before this spectacular plunge we had witnessed several other comets graze (come close without hitting) the Sun.  

I've included a nice picture of comet Lovejoy after its close approach of the Sun brought it through the Sun's corona around December 15, 2011:



What happens when a comet hits the sun?

It acts like a supersonic snowball in Hell


  If a comet is big enough and passes close enough, the steep fall into
  the sun’s gravity would accelerate it to more than 600 kilometres per
  second. At that speed, drag from the sun’s lower atmosphere would
  flatten the comet into a pancake right before it exploded in an
  airburst, releasing ultraviolet radiation and X-rays that we could see
  with modern instruments.
  
  The crash would unleash as much energy as a magnetic flare or coronal
  mass ejection, but over a much smaller area. “It’s like a bomb being
  released in the sun’s atmosphere,” Brown says. The momentum propelled
  by the comet could even make the sun ring like a bell with subsequent
  sun-quakes echoing through the solar atmosphere.