Hiei from YuYu Hakusho has two attacks that are comprised mainly of hellfire. The first is Fist of the Mortal Flame, which uses the fire from human hell to pummel the opponent. The other, much more dangerous attack is Dragon of the Darkness Flame, which is an enormous dragon made of flames from the demon hell.
In Blue Exorcist, blue flames are the sign of Satan. Rin, a son of Satan, tends to erupt into them. This is a problem due to Fantastic Racism and also a boon; there isn't much that can handle them after they are the only demonic flames that attack the physical and the spiritual.
Shima can summon a demon that uses black hellfire.
The dragon Atlas Flame describes his flames as hellfire. Natsu tries to eat him too for a power boost. It might have worked too, if Natsu and Atlas Flame hadn't instead joined forces once they realized they were both close to Igneel.
Then of course there's the actual Hellfire card, which wipes out all non-black creatures (being a black spell, it's assumed that most if not all of your own creatures will be exempt) at a price.
Fantastic Four example: Doctor Doom opens a portal to Hell, and Johnny and Franklin are pulled inside. Johnny gets out, but not without being badly burned; especially notable considering he's the Human Torch, master of flame in all forms, who hasn't suffered a burn in, like, thirteen years. Yeah. Hellfire. It'll mess you up.
Though Johnny later finds out that the holy flames of an archangel's Flaming Sword are even more painful.
In the Betsy the Vampire Queen books by Mary Janice Davidson, the titular queen's half sister Laura is the daughter of Satan (don't ask). As such, one of Laura's powers is to summon a sword made of hellfire. It can transform into a crossbow in the blink of an eye, and is implied to always be hanging at Laura's hip, invisible when she doesn't need it. Hellfire only disrupts magic, so it passes harmlessly through mundane humans but incinerates vampires instantly. However, Betsy's odd status as Queen of the Dead means that the sword neither passes harmlessly through nor incinerates her, but stabs her like a normal sword would. It gets stuck and must be pulled out, but leaves no wound behind.
In Dungeons & Dragons, it's manufactured by Devils and can burn creatures made of fire. Ever watch a fire elemental burn to death? Not pretty Absolutely awesome. (Some sources say that it's "not truly fire" at all" but an evil form of energy unique to Hell.) Particularly useful since about half the monsters in the game are resistant to normal fire to some extent.
Third edition has Mephistopheles, the Lord of Hellfire, who grants warlock followers the power to draw upon his powers of hellfire that function as normal fire damage, except that effects that provide resistance or immunity to fire damage don't affect them. He claims to have discovered Hellfire; whether this is true or not, he is, no doubt, better at using it than any other devil, and has created many horrible devices associated with it, such as Hellfire Engines.
Hellfire Warlocks are a type of mortal wizard who can harness this form of energy, mentioned in the 3.5 Edition. (Not all of them are evil, but most are.)
In an interesting variation, infernal powers in Exalted tend to use the burning, poisonous light of Ligier, the demonic Green Sun of Malfeas, in the same manner as characters in other settings use hellfire. In addition to the usual attributes of a Hellfire-analogue, this can also infect victims with Green Sun Wasting, a truly horrifying supernatural disease that makes Ebola look like the common cold in comparison.
Hellfire is a damage type in the roguelike game TOME. Unlike normal fire, it cannot be resisted and evil creatures take double damage. The game also has holy fire.
Similarly, the Roguelike crawl has a damage type hellfire - some monsters resist it, players can't (contrary to almost all other types of damage). Hellfire attacks are always ranged area effects. They're most commonly used by various demonic monsters at the high end of the power scale, but can also be invoked by player characters with a specific mutation or wielding one of a very small set of artifacts.
World of Warcraft has this as a - surprise! - warlock spell, which spews apparently unholy flame in a radius around the warlock, dealing decent damage per second in an aoe but also injures the warlock. Oddly enough, the self-damaging part is the thing warlocks use it for as they have more powerful AoE but dying to hellfire doesn't cause durability loss.
City of Heroes soon-to-be-released Demon Summoning powerset for Masterminds has a few demons that attack with hellfire, as well as giving the Mastermind a whip made of it. While regular fire attacks have the secondary effect of igniting enemies for additional fire-type damage over time, hellfire's damage over time is toxic-type, and applies a damage resistance debuff to its targets.
Scorpion of Mortal Kombat fame has power over hellfire as a side effect of being undead. He pretty much only uses it to kill people by breathing it on them. One of his special moves from Mortal Kombat: Deadly Alliance onward lets him summon flames from the ground.
Hellfire is an attack in Bravely Default that is appropriately used by Cerberuses, which deals 1.25 fire damage to all enemies. You can utilise the attack yourself by absorbing it with the Vampire's Genome Drain.
The titular character of the webcomic Zebra Girl has the ability to summon hellfire which makes anyone she's mad at spontaneously combust. In a subversion, Epileptic Tree wisdom states that the burns made are skin deep, so that the victim can be fried again...and again...and again.
Flames from other supernatural sources
In Slayers, there's a distinction: for example, Fireball is Shamanism spell — it calls upon fire spirit. Gaav Flare, on the other hand, was Black Magic using the power of Chaos Dragon, one of Dark Lords. Defunct after Gaav's demise, of course. Reflecting Gaav's nature and power level, it's much nastier and burns through the first target, then whoever was behind it...
Thirty Hs grants Harry Potter the power to wreathe his fists in "Holy Fuckfire" with which he punches the heads off vampires, sending them into the past of Mars, or... something. Notable for being evidently a holy force rather than an infernal one, and... powered by swearing, or something.
Harry Dresden of The Dresden Files': After demonic shadow who gives him access to hellfire dies saving Harry's life he finds out that the archangel in charge of maintaining free will, and God's own holy hitman, Urielnote The one who killed all of Egypt's firstborns has gifted him with Soulfire, the fires of Creation itself. It fits the trope just as well as normal Hellfire - it makes the spells more "real," and thus more powerful, but at the cost of some of Harry's soul (though it regenerates so long as he has some left. It regenerates more quickly if he takes time to nurture it; for example, going on a date with Luccio at the end of Small Favor''). The Soulfire also burns an unusual color - it usually adds a bit of a silver tinge, depending on how much he uses and what spell he ties it to.
Journey to Chaos: Kasile possesses Sacred Fire which comes from the fire goddess Fiol. It is white in color, and unlike regular fire or magic fire, this stuff is divine in nature and not bound by the same limitations. For instance, it can't be blocked by Orichalcum.
Dungeons & Dragons: The cleric spell flame strike has shades of this, attacking with a 50-50 mix of fire and divine energy. The fire can be resisted, but the other half can't.
Referred to as 'Wizard's Fire' in Sword of Truth. It's described somewhat like a magical version of napalm, a "liquid flame" which won't be put out by smothering, even if it's just a little bit of flame; in fact, it will just set on fire whatever you use to smother it. Beyond that, an even stronger version known as Wizard's Life Fire appears at a few points, used as a Desperation Attack by wizards who are about to die anyway.
A Song of Ice and Fire has wildfire and dragon's breath. Wildfire is the exclusive creation of the Alchemist's Guild, a liquid something like magical napalm which burns with an intense green flame that's impossible to extinguish (it can burn water!), and seems to be almost alive in larger conflagrations. It seems to be based on Greek fire. Dragon's breath is apparently much hotter than normal flames, and is said to possess magical qualities, such as its use in the creation of Valyrian Steel.
Science Fiction variant in Charles Stross' Glasshouse - "Blasters" are very simple weapons based around a couple of wormholes. One end opens at the end of your pistol; the other opens into a sun.
For a less evil version, the Searing Spell feat from Sandstorm can make your normal fire spells burn hot enough to deal partial damage even to creatures immune to fire.
Pokemon: Black and White also gives us the inaccurate but powerful Inferno, which will always leave a burn. To further the resemblance, it's actually called Purgatory in Japan.
Other abnormal flames
Theories that it summons fire from hell aside, Amaterasu from Naruto doesn't have any connection to hell, but exhibits many traits as hell fire: special color (black), extremely hot (supposedly as hot as the sun), has impossible properties (can burn non-flammable objects, even underwater & can cause even Bijuus who shrug off lesser attacks to cry out in pain), and is very hard to put out (can't be put out unless you wait for a week, seal it, or the user snuffs it out).
Zetsu actually says at one point that Amateratsu is supposedly "black hellfire". It's real nature is never clarified.
The tailed beasts' chakra isn't really fire of any sort, but when a Tailed Beast's host put out enough of the chakra, it can incinerate anything it touches... even the host himself.
Zancrow summons and controls pitch-black flames that can burn and injure Natsu. Natsu is normally immune to heat and eats flames. Natsu eventually manages to eat Zancrow's flames, giving him the power boost to defeat him.
Poké Wars depicts Will-O-Wisp in this form. It is a sinister dark purple/blue flame that ignites anything it touches and cannot be extinguished.
In Harmony Theory, Nightmare Umbra controls Ashfire. One touch and it consumes you until nothing is left but ash, and it is almost impossible to extinguish. It feeds on magic. Star Fall is able to survive it by redirecting her internal magic away from it, so it goes out from lack of fuel.
While not specifically from hell, fiendfyre in Harry Potter is hot enough to destroy a horcrux, something which is rather difficult to do. And is sentient. And malevolent. And a lot harder to put out than to start. And very difficult for a normal wizard to control, so it's not like you can practice with it.
''The Dresden Files': There's also Summer Fire, a gift granted by the Summer Court. So far, though, it's only been used against its polar opposite in the form of the Winter Fortress, where it is understandably extremely destructive, so no word on if it's actually any hotter or more powerful (though it does leave a trace in the user's fire magic from then on).
A certain spell in Chronicles Of The Raven makes very powerful jets of fire that supposedly home in on enemies. Have a guess at its name.
The Banned and the Banished also uses the term "balefire" for fire from evil magic. It has many properties of normal fire, but it freezes instead of burns. (Good spellcasters can also create fire, but once started it functions like normal fire.)
The destructive, rare and difficult to control white fire from The Chronicles of Thomas Covenant. Curiously, "Hellfire!" is one of Thomas Covenant's favorite sayings.
Adapted in a third-party setting called "Infernum". There, it's basically negative spiritual energy (despair, pain, misery) given physical expression as sickly green-black flames, which are capable of consuming flesh, bone and soul with equal ease and which thus makes it especially powerful against creatures like demons and angels. It consequently has its own damage type (and damage resistance), so ordinary Fire resistance is worthless against it (although, conversely, a character with only Hellfire Resistance is defenseless against Fire damage).
Warpfire in Warhammer and Warhammer 40,000 is evil fire drawn from the Warp. Unsurprisingly, it's chiefly the weapon of daemons, their servants, and the odd race of evil rat-men (who have Warpfire flamethrowers). Tzeentch's daemons in particular are little more than living flamethrowers, and vehicles dedicated to Tzeentch in 40k are covered in eldritch fire. There's also the Holocaust power, which is incredibly dangerous and deadly, and burns Demons just as well as anything. Bonus; it makes resurrection impossible and even kills immortal demons.
In the Exile/Avernum series, quickfire is an artificial, very powerful magic flame that ignites even thin air to create a wall of flames that spreads as fast as a man can run. Nothing short of cold rock or very powerful magic can stop its spread.
Chaos Bolt, another warlock spell, is probably a better example. It fires a bolt of chaotic fire that goes right through absorption effects and ignores fire resistance. It does not, however, work against targets that are completely immune to fire damage.
WotLK also introduced the concept of dual-element spells, primarily so you can't easily resist them by stacking a specific magic resistance (usually fire). Spellfire, Spellfrost, Frostfire, etc.
Castlevania's Dracula has a three-to-five-directional fireball attack by this name. On occasion, he cranks up the damage factor and throws big black METEOR-fireballs at you.
Alucard can mimic the three-fireball and two-meteor-fireball attacks EXACTLY in Castlevania: Symphony of the Night. It is so immensely satisfying to be able to pull Dracula's shenanigans on his henchtwits. To gamers unfamiliar with the old Nintendo Hard console titles and their relatively slow Belmont heroes, the fireballs may seem like small potatoes, but even very skilled, hardcore gamers have been reduced to incoherent howling by the original Castlevania and the Sharp X68000 remake of that title. (We do not mention Castlevania III: Dracula's Curse around these gamers. They will go insane.)
Soma Cruz, can also perform the three-fireball attack if he equips the right soul. In Castlevania: Aria of Sorrow, it's one of the three souls he needs to equip during the battle with the Disc-One Final Boss to unlock the path to the true ending.
A more serious version is Shadow Fire from XD. Exclusive to Moltres, it's a Shadow-empowered Flamethrower attack that will barbecue even the sturdiest Water, Rock, and Dragon Pokemon. Then again, Shadow power is super-effective against everything other than itself...
The Gold Pokedex entry for Houndoom is meant to evoke this trope; apparently, the pain inflicted from the flames it breathes will never go away.
Dragon Cave has a species called "Hellfire Wyverns". The submissive males are the bright fiery red usually associated with this trope, while the more violent females are the bright blue of hotter flames.
Other (not an example?)
The science fiction novel Roadside Picnic mentions a substance called "Witch's Jelly", left behind by the mysterious alien visitors, that will burn (or corrode?) through just about anything, leaving a flaming pit in the ground as it goes.