Personally, I am ambivalent. If I remember to turn off the kettle before it boils, I'll do it to save time, but I often forget.
That said, some people are absolutely vehement about their methods.
The boil-first side likely gets their argument from tradition, when boiling water was needed to make it safe, but it is now a moot point in most of the Western world. The no-boil side usually claims that boiling removes oxygen; I don't know if this is true, but maybe. Water that has been boiled over and over again does taste a bit flat to me, but I don't think one boil will hurt. Then, sometimes people will bring qi into the equation, at which point I become completely lost.
(Of course, not all people who boil/don't boil think along these lines, these are just the classic arguments when people debate the issue)