[Thingol]

There are 2 reasons why I do not believe that the secret fire is the ring Narya; which is what Illuvitar hid from Melkor, Luineglin was correct. The first is some textual evidence from the Silmarilion.

"He [Melkor] had gone often alone into the void places seeking the Imperishable Flame; for desire grew hot within him to bring into Being things of his own, and it seemed to him that Iluvatar took no thought for the Void, and he was impatient of its emptiness. Yet he found not the Fire, for it is with Iluvatar."

Then later in the text:
"Therefore Iluvatar gave to their vision Being, and set it amid the Void, and the Secret Fire was sent to burn at the heart of the World; and it was called Ea."
(Silmarillion)

The Imperishable Flame and the Secret Fire seem to be the same thing; the power of Iluvatar to impart actual Being to his thought. It seems fitting that this would be something that Gandalf (as a Maia sent from the West by the Valar) would "serve" and by contrast it does not seem likely that Gandalf would serve his ring Narya. However, I'm still unclear on what the flame of Arnor is. Gandalf says he is a wielder of the flame of Arnor, and I do not think he is a wielder of the sun =).