I just acquired the Lordvessel and placed it on the Firelink Altar. When I did, a cutscene showed 3 orange barriers opening. Where can I find those barriers/doors, and what areas do they lead to?
There are several orange fog gates in the game which can only be lifted by obtaining and placing the Lordvessel at the Altar. These fog gates can be found in Anor Londo, Tomb of the Giants and Demon Ruins.
Tomb of the giants (past catacombs): The fog gate blocks the way to the boss.
Anor Lando: Fog gate is on boundary between Anor Lando and the Dukes Archives (preventing access to the archives). Exit left from the Anor lando bonfire or head straight from where you enter from Sen's Fortress
Demon Runes: Fog gate is in demon ruins and blocks your access to the demon firesage boss and Lost Izalith. The fog gate is in the area of the demon ruins that is only accessible once you defeat the ceaseless discharge boss.