The unnamed fire Avatar first appeared to Aang in a spiritual vision directed by Avatar Roku, in which he was lavabending the magma from several volcanoes while in the Avatar State. The young airbender saw him again briefly in a similar vision during the battle at Wulong Forest, this time among a great number of Avatars all standing in a line, upon regaining his ability to enter the Avatar State.
Aang temporarily lost his connection to this fire Avatar after removing the firebending emblem from his meditation beads; however, upon crafting and attaching a new one, he saw him again in a lineup of his past lives to indicate his restored link.
Appearances in Korra's life
This Avatar was present, along with all the other Avatars, when Aang passed on the knowledge of energybending to Korra and restored her bending, representing the newly established connection she had made with her spiritual self.
After Unalaq had merged with Vaatu and proceeded to attack Raava, the fire Avatar was present in the Avatar lineup that Korra envisioned; however, his connection to the latter was severed due to Unalaq's assault on and subsequent destruction of the Light Spirit.