Not completely true.
Fire elementals do not have an aura which causes fire damage, though things they attack do take extra fire damage. However, possessing something does not cause fire damage because the rules do not say they do.
While it might be logical to say it happens, the rules don't.
Also metal weapons and armor and likely immune even if you said burn damage applied. Weapon and armor all have hardness, and magic enhances it.
Energy Attacks: Energy attacks deal half damage to most objects. Divide the damage by 2 before applying the object's hardness. Some energy types might be particularly effective against certain objects, subject to GM discretion. For example, fire might do full damage against parchment, cloth, and other objects that burn easily. Sonic might do full damage against glass and crystal objects.
Hardness: Each object has hardness—a number that represents how well it resists damage. When an object is damaged, subtract its hardness from the damage. Only damage in excess of its hardness is deducted from the object's hit points (see Table: Common Armor, Weapon, and Shield Hardness and Hit Points, Table: Substance Hardness and Hit Points, and Table: Object Hardness and Hit Points).
Firstly energy damage is reduced by half against object. Then you apply hardness. An elder giant fire elemental has a burn which deals 2d10 damage. Which would be halved and then you would subtract 10 for metal objects. Even if you rolled maximum damage for 20 fire damage on a burn, once reduced by half and then subtracting hardness means 0 damage.
It is never stated to, so it doesn't.
However, bear in mind the area of affect spells or abilities do not actually cause damage to gear, but creatures only. However a targetted ability could target a piece of gear and cause energy damage to it. It's just more common in a game to see AoE energy atatcks than targetted.