The Huo Chien, or Fire Arrow, was a projectile weapon. It was the Special Weapon of Sun Tzu. It was also featured as one of the arrow types used by Genghis Khan. Vikings have also used flamming arrows.
The Fire Arrow was a relatively simple weapon. The arrow head was wrapped in cloth and dipped in sesame oil, and then covered in lard and was ignited prior to firing.
The Art of War lists five methods of using fire in an attack: against an enemy camp, against enemy stores, against supply lines, against arsenals, and against enemy soldiers in the field. It was mostly used during dry seasons, when the fire would have spread the most quickly.
Historically; fire arrows were specifically used to burn buildings and towns instead of against humans. As the show pointed out; the penetration and lethality of the arrows are weaker compared to ordinary arrows.
Fire Arrows did see use in naval warfare since ancient ships were mostly wood. Fire Arrows were also aimed at sails to slow the ship down. Greek Fire and similar proto-napalm became more effective during the medieval age.
Incans would use slings with hot stones alongside fire arrows. This saw success during the Siege of Cusco; where Manco Inca almost managed to burn a building Hernando Pizarro was hiding in. Hernando's slaves however saved him by putting out the fire with buckets of water; despite being exposed to the enemy Incans. Conquistadors claimed that Saint Mary descended from heaven to extinguish the flames.
Ninja used explosives to cause more damage with their arrows: these weapons were called Ookuni-Hiya. These explosive arrows did see in other regions of the world, but were overall uncommon as musket technology outperformed these arrows in terms of effective range.
Other Examples of Fire Warfare
In the 1187 Battle of Hattin, the Abbasid Caliphate led by Saladin managed to calculate the direction of the wind and used that to burn nearby trees to create a cloud of smoke to engulf crusading knights. While this tactic wasn't lethal, it did weaken the already exhausted and dehydrated knights.
The Huo Chien was tested against the hand cannon. When an area was prepared for it's use it was shown to be highly effective, however when used against a single opponent it proven to be highly ineffective. The wrappings around the arrow head keep the arrow from penetrating deeply enough to actually cause serious harm, and the force of the impact caused the flame to go out. It also proved to be counter-productive, as the heated arrow tip cauterized the small wound it did inflict, making it bleed less.