While projection mapping has recently exploded into the conciousness of artists and advertisers everywhere, the history of projection mapping dates back longer than you may imagine.
If you try Googling for "Projection Mapping" you won't find anything older than 3 years. That is because projection mapping's older, academic name is "Spatial Augmented Reality" . The field is also known as "video mapping", but projection mapping seems to be winning out in the United States.
For the purposes of this history, I'm only including work that considered projection onto an arbitrarily complex surfaces. Projection onto flat and cylindrical/spherical surfaces has a much older history and goes back to the invention of cinema.
The first known instance of projection onto a non-flat surface dates back to the 1969 opening of the Haunted Mansion ride in Disneyland. The dark ride featured a number of interesting optical illusions, including a disembodied head, Madame Leota, and 5 singing busts, the 'Grim Grinning Ghosts', singing the theme song of the ride. These were accomplished by filming head-shots of the singers (with 16 mm film) and then projecting this film onto busts of their faces.
The next projection mapping instance comes in 1980, with the immersive film installation Displacements by Michael Naimark. In this art installation a living room with two performers were filmed with a rotating camera, then the camera was replaced with a projector. The result is rotating projection mapping.
Disney not only pioneered the technology of projection mapping, they also have the earliest patent (that I can find) in the space. Entitled "Apparatus and method for projection upon a three-dimensional object". It essentially describes a system for digitally painting an image onto "a countoured, three-dimensional object."
GE also has an early patent for a "A system and method for precisely superimposing images of computer models in three-dimensional space to a corresponding physical object in physical space."
Projection mapping really started to get traction when it was pursued in academia. "Spatial Augmented Reality" was born out of the work by at UNC Chapel Hill by Ramesh Raskar, Greg Welch, Henry Fuchs and Deepak Bandyopadhyay et al. It all got started with a paper The Office of the Future .
The Office of the Future envisioned a world where projectors could cover any surface. Instead of staring at a small computer monitor, we would be able to experience augmented reality right from our desk. This means we could Skype with life-size versions of our office mates, view life-size virtual 3D models. This work even featured an early real-time, imperceptible 3D scanner (like the Kinect).
A year later, UNC defined spatial augmented reality, demonstrating the basics of projecting textures onto 3D objects .
You may know John Underkoffler as the designer who invented the Minority Report interface, and the Chief Scientist of Oblong Industries, Inc. But before that, he pioneered some of the early work in interactive projection mapping.
He introduced the concept of the I/O Bulb (Input/Output Bulb), namely a projector coupled with a camera that could one be as ubiqitous as a traditional light bulb.
Then in 2001, Raskar's follow on Shader Lamps work.
From there Raskar et al. went on to explore moveable projectors (predicting the pico projectors of the future) . These hand-held smart-projectors are aware of their position and orientation through a variety of sensors. They demonstrated using smart projectors to aid in warehouse inventory and maintenance.
Oliver Bimber then explored projecting onto paintings  and converting drapes into projection screens .
 Bimber, Oliver, Ramesh Raskar, and Masahiko Inami. Spatial augmented reality. AK Peters, 2005.
 Raskar, Ramesh, et al. "The office of the future: A unified approach to image-based modeling and spatially immersive displays." Proceedings of the 25th annual conference on Computer graphics and interactive techniques. ACM, 1998.
 Raskar, Ramesh, Greg Welch, and Wei-Chao Chen. "Table-top spatially-augmented realty: bringing physical models to life with projected imagery."Augmented Reality, 1999. (IWAR'99)
 Raskar, Ramesh, et al. "Shader lamps: Animating real objects with image-based illumination." Proceedings of the 12th Eurographics Workshop on Rendering Techniques. 2001.
 Bandyopadhyay, Deepak, Ramesh Raskar, and Henry Fuchs. "Dynamic shader lamps: Painting on movable objects." Augmented Reality, 2001. Proceedings. IEEE and ACM International Symposium on. IEEE, 2001.
 Raskar, Ramesh, et al. "RFIG: interacting with a self-describing world via photosensing wireless tags and projectors." ACM Transactions on Graphics (TOG). Vol. 23. No. 3. ACM, 2004.
 Bimber, Oliver, et al. "Superimposing pictorial artwork with projected imagery."ACM SIGGRAPH 2006 Courses. ACM, 2006.
 Bimber, Oliver, Andreas Emmerling, and Thomas Klemmer. "Embedded entertainment with smart projectors." Computer 38.1 (2005): 48-55.