The main phenomenon you are encountering here is the so-called dativus commodi or Dativ des Vorteils which is a rather rare construction in German that indicates to whose advantage or disadvantage (dativus incommodi or Dativ des Nachteils) something is happening, or more generally to whom something is happening. This is a free dative that is not related to a certain verb requiring this case.
A few simple examples:
Der Krug zerbrach ihm.
literally: The jug broke and this happened to him.
freely: He (unintentionally) broke the jug.
Er putzte ihr die Schuhe.
literally: He cleaned the shoes for her.
freely: He cleaned her shoes.
Now let’s look at your sentence and add sentence parts one-by-one:
Eine Stimme fällt.
A voice falls.
Eine Stimme fällt vom Gesicht.
A voice falls from the face.
Eine Stimme fällt dem Himmel vom Gesicht.
literally A: A voice falls from the face to the sky’s disadvantage.
literally B: A voice falls from the face and this is happening to the sky.
freely: A voice falls from the sky’s face.
Eine Stimme aus dem Licht fällt dem Himmel vom Gesicht.
A voice from the light falls from the sky’s face.
As already noted by O.R. Mapper, switching the order of dem Himmel and vom Gesicht would not change the meaning, but result in an unusual or marked sentence structure, emphasising Himmel.