MovieTexture is finally deprecated after Unity 5.6.0b1 release and new API that plays video on both Desktop and Mobile devices is now released.
VideoPlayer and Video
I used @Programmer 's answer to play videos from a URL, but I couldn't get any sound to play. Eventually I found the answer in the comments of a YouTube tutorial.
To get the audio to play for a movie loaded via URL, you need to add the following line before the call to EnableAudioTrack
:
videoPlayer.controlledAudioTrackCount = 1;