What was the deal with the amulet Aang burns at the end? Where did he get it from? How does it have a connection to his past lives? Do the all do that, or just the fire one with Roku? Am I missing something, or is it never explained?
I believe those were his meditation beads. It seems like something monks and senior Air Nomads used, since it was part of his garb during Zuko's coronation. I don't think they have a connection to his past lives, rather they focus his energy during meditation so in essence he can channel a life (much like how he projected past Avatars when he was on the lion turtle). The fire amulet symbolized his connection to Roku, so when he burned it, it corresponded to his severing of his connection to Roku.
It was probably a mental/spiritual sort of thing. Aang seems to have had something of an open door to his past lives, likely due to his spiritual nature (as opposed to Korra's door sealed with caulk), especially for Roku. After he realized that Roku could bring nothing new to the table, he probably mentally/spiritually blocked Roku from manifesting to him like he did in the original series through his own willpower. Sort of a sign of his greater control over his Avatar nature. As for the amulet, it was likely burned for the same reason Zuko and Iroh cut off their topknots: it was a physical representation of something that Aang wanted to internally cast aside, a reminder of Roku's constant presence within Aang.