Dysfunctional Rules X: I Cast Comprehend Rules

[Gruftzwerg]

Unfortunately, this doesn't actually resolve the dysfunction. The fire elemental's burn ability only says that the elemental's slam attack deals fire damage and sets people on fire. Other attacks don't do so.

Burn (Ex)

A fire elemental’s slam attack deals bludgeoning damage plus fire damage from the elemental’s flaming body. Those hit by a fire elemental’s slam attack also must succeed on a Reflex save or catch on fire. The flame burns for 1d4 rounds. The save DC varies with the elemental’s size (see table). A burning creature can take a move action to put out the flame. The save DC is Constitution-based.

Creatures hitting a fire elemental with natural weapons or unarmed attacks take fire damage as though hit by the elemental’s attack, and also catch on fire unless they succeed on a Reflex save.

"Unarmed Attacks" doesn't sole apply to "Unarmed Strikes" but also to "Grapple".

Unarmed Attacks != Unarmed Strike

Unarmed Attacks = Unarmed Strike; Grapple;...

edit: and while the elemental does not burn an enemy for "grappling" it itself, it will use its slam attack, thus still apply his burn ability.