Fire is hot—except for those occasions when it's not. Sometimes, one of the special properties of magical or illusory fire is that it doesn't burn things, although, oddly enough, it may still require fuel. Sometimes, the non-burning property only applies to certain classes of objects, such as fire which doesn't burn people but will burn anything else, or fire which only burns living organisms.
Sometimes, the flames will actually be cold (in varying intensities), and not just not hot (scientifically a flame that's cold wouldn't be flame, but rather a cryogenic vapor that's luminescent, causing it to look like flame).
See also Convection Schmonvection, for when fires supposedly are hot, but don't quite behave like it.
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Anime and Manga
In One Piece, Word of God states that Marco's blue phoenix flames do not burn or produce heat like normal flames, instead they are what allow him to heal.
One Jack 'O Lantern (who later became Hobgoblin), a recurring Spider-Man villain, has flames all over his helmet, yet doesn't get burned due to the fire being a variant called "stage fire," which is used in theatrical productions to simulate flame. He also has an air-supply within the helmet — separate from the fire's air supply — that helps keep the helmet cool.
In Artemis Fowl: The Time Paradox, Damon Kronski, leader of the Extinctionists, has a pit lined with flamethrowers which he uses to stage the execution of endangered species for the amusement of his supporters. Unbeknownst to him, another character (who wants the endangered animals saved for selfish reasons) fitted a trapdoor in the base of the pit and replaced the flamethrowers with hologram generators.
In the Coldfire Trilogy, the Anti-Villain developed his own version of fire magic that burns cold rather than hot. As an undead, it's the only way he can safely wield fire.
In Susan Cooper's The Dark Is Rising, Hawkin summons nine large magical freezing cold flames that represent the power of the Dark and the spells of the Deep Cold.
An entire novel in The Galactic Consul series by Yevgeny Filenko is set on a planet whose native culture discovered a form of low-temperature plasma that doesn't burn if handled right (but is deadly otherwise) and adapted it into a weapon. The scientists are as baffled by its physics by the end of the novel as they are at the start.
In Harry Potter and the Philosopher's Stone, Hermione puts some non-burning magical fire in a jar to keep the trio warm in winter. Later she uses it to trick Snape into thinking his robes have caught fire and scoops it back into the jar when she's done. (In the film, the latter is changed to the regular kind of fire.)
Later on in book, during the trials to reach the Stone, Snape's challenge has the entrance and exit blocked off by magical fire. Only by drinking the correct potion will it be safe for one to pass through them.
In Harry Potter and the Prisoner of Azkaban, the textbook Harry reads reveals that the spell to negate flames is called the Flame Freezing Charm, and reduces their effects to a mild tickling sensation. During the era of witch burnings, witches and wizards would use this charm to survive their execution and subsequently fake their deaths. One witch, Wendelin the Weird, loved being "burned" and let herself be captured several times in various disguises.
The Saga of Hervor and Heidrek: When Hervor is looking for her father's gravemound in a haunted burial ground at night, there are ghostly "grave-fires" burning on the barrows. As it turns out, Hervor can go right through these fires without getting burned.
Minor Outer Gods Aphoom-Zhah and Tulzscha from the Cthulhu Mythos are both Elemental Embodiments of this sort. The former is a gray flame that consumes heat rather than emitting it, whilst the latter is comprised of "sickly green" flames that "radiate the clamminess of death and corruption" and leave a "noxious, venomous verdigris" wherever they touch. Aphoom-Zhah literally freezes whatever it touches, though Tulzscha is implied to be more of a poison or radiation elemental.
Mythology and Religion
In the Book of Exodus, an angel of the Lord speaks to Moses from a bush that burns but is not consumed.
In Judaism, mainly kabbalah, there are 4 types of flames that are differentiated by their capabilities:
Fire that consumes solids but does not vaporize water
Fire that does not consume solids but does vaporize water
Fire that consumes solids and that also consumes water
Fire that doesn't consume solids or vaporize water
Module I4 Oasis of the White Palm. In the Temple of Set the PCs can find a brazier filled with violet flames. The flames don't give off heat and don't burn wood. However, if they touch living flesh, they burn it, causing serious damage.
The Continual Flame spell creates a permanent fire that doesn't burn or use oxygen and is used to make Everlasting Torches.
In the Mystara D&D setting and the CD&D game, the halfling Masters' racial artifact is blackflame: a dark-colored, frigid "fire" that burns inflammable substances and radiates shadow rather than light.
In Call of Duty multiplayer maps, fire is often present in small amounts (think burning wreckage) but entirely cosmetic. A player can stand next to or even in it and suffer no damage.
Subverted with the Frozen Flame in Chrono Cross. While there are many different in-universe ideas of what this thing is, most people simply think it's some kind of cold fire, or fire trapped in ice (That has special powers). It's actually a fragment of Lavos that survived its destruction after the events of the previous game.
The seventh The Land Before Time sequel is subtitled "The Stone of Cold Fire", referring to a comet seen crashing by Littlefoot. A few characters believe it to be a legendary "Stone of Cold Fire" that would grant their finders special powers. When found, it just turns out to be a mere rock, though.