What is the lighting technology behind the Dreamliner's dynamic interior lighting?
This 'mood' lighting is offered by Boeing in 787 and Airbus in A350XWB (as an option) and is achieved by using Light Emitting Diodes (LED). According to Airbus,


  Airbus is applying the latest technology in energy-efficient LED (light-emitting diode) mood-lighting on the A350 XWB. This enables the cabin ambience to be tailored to each airline’s own identity, as well as mimicking natural sunrise and sunset illumination that helps the body adapt to jetlag on long-range flights.


The main advantages of LEDs over conventional lights are,


The LEDs are longer lasting (~50,000 hrs).
They are more energy efficient.
They generate less heat.


Phillips seems to be involved in the development of lighting systems for the(se) aircraft. According to led-professional.com,


  Philips Lumileds provided the LEDs while Diehl Aerospace engineered the overall lighting system for the Boeing and Airbus cabin interiors. 


The light intensity and colors are decided by the airliners and are pre-programmable (and can be changed remotely). According to Bloomberg,


  The Finnair A350 cabin has two dozen light settings, aligned with stages of a long-haul flight. It will also feature warmer, amber colors on flights arriving in Asia and cooler “Nordic blue” hues when flying into Finland. ... 
  Virgin Atlantic has five primary color periods on its 787 flights: rose-champagne for boarding, purple-pink for drinks, "amber" for dining, another for the pre-sleep period called “work-rest-play,” a silvery glow for overnight sleep, and a waking color. 


Similar lighting systems are also offered by third parties for other aircraft.