The hellhound is naturally impervious to fire itself, even in their human guises, though the human side still feels the pain of fire before they can trigger their true nature.
Deputy Jordan Parrish was enveloped in the explosion of a makeshift bomb but survived walking away, unscathed. In Perishable, he was soaked in gasoline then set aflame: his entire frame was completely unharmed, even his hair and fingernails.
A hellhound wields pyrokinesis: they can summon, generate flames or extreme heat energy from within their own body of their own will without harming themselves.
Fire also seems to be a part of the hellhound's physiology. In Creatures of the Night, Parrish gave off smoke from his wounds whilst healing from a massive injury. When Parrish shows his 'Beta' hellhound shape: his body displays fissures secreting lava upon his skin that appear like cracks on a volcano