"The Funeral of a Viking" (Frank B. Dicksee, 1893)

A subtrope of Burial At Sea in which the deceased (usually a warrior, but not necessarily a Viking) is laid in a boat with his effects, such as a sword, and set out to drift at sea. Then, a character (usually someone with emotional ties to the dead, or simply the most skilled Archer present) lights a flaming arrow and shoots it at the sail. The ship is then engulfed in flame and slowly breaks up and sinks.

Played straight, it can be a Tear Jerker. When Played for Laughs, the character charged with shooting the flaming arrow will continually miss or fumble his bow.^*

In some cases, the fire-arrow step isn't performed. This can be due to a lack of flammable sails and fire arrows, a need for haste, or simply different funerary practices.

A third variant exists in which the boat is not set out to sea, but instead used to lay the corpse in for a funeral pyre.

And finally, Viking funerals need not be literal. Many examples feature metaphorical stand-ins for any of the aspects of the ritual.

Because this is a Death Trope, there are unmarked spoilers ahead. Read at your own risk.

Examples:

Anime and Manga

On One Piece, the Straw Hat Pirates do this for the Going Merry, their first ship.
In chapter 39 of Vinland Saga, a Viking Funeral is held for Thors.

Comic Books

Viking funerals are common amongst the Asgardians of the Marvel Universe. They have even been known to extend the practice to outsiders who prove themselves worthy. When Asgard was floating over Oklahoma (It Makes Sense in Context) they made funeral pyres with boats.
The eponymous V from V for Vendetta is given a funeral in the spirit of this trope, with the train acting as a proxy for the traditional boat. Instead of being shot with a fire arrow, the train is packed with explosives.
Lauren in DMZ is given one of these, although instead of a wooden boat and a flaming arrow it's an inflatable raft and a flare gun.

Film

Happens in the Hollywood epic The Vikings (1958) by Richard Fleischer. The Viking Funeral of Viking prince Einar (Kirk Douglas) is the last scene of the film (see it on YouTube). For cinema, this is certainly the Trope Codifier.
King Arthur is laid to rest this way in First Knight.
The Robert Zemeckis Beowulf film features one for Beowulf himself.
History of the World Part I features such a funeral in the Coming Attractions segment.
A Viking funeral pyre is the standard funerary procedure for Jedi in the Star Wars films, the most iconic being Darth Vader's at the end of Return of the Jedi.
Happens at the end of Outlander. A similar funeral at the beginning of the film would have made Book Ends, but was cut for time.
It Runs In The Family (2003) features a Viking funeral that is both a Tear Jerker and one of the funniest scenes in the film, particularly because it is set in modern day New York City.
Ray the firefly's funeral at the end of The Princess and the Frog. Shortly after that, he comes back as a star in the sky.
King Harold is actually given a funeral resembling this after croaking at the very beginning of ''Shrek the Third''. It even comes with a choir of singing frogs performing his funeral dirge.
In the Conan movie, Conan burns Valeria on a funeral pyre after she's killed by Thulsa Doom. It's especially poignant for Subotai's line.
Subotai: He is Conan, Cimmerian. He will not cry, so I cry for him.
In S.O.B., the heroes decide their friend needs a better send-off than a Hollywood funeral full of phonies so they steal his body from the funeral home and send it out to sea in a burning boat.
In Van Helsing, the titular character gives the love interest this treatment after accidentally killing her.
The grandfather in Rocket Gibraltar.
The film Grand Theft Parsons revolves around the protagonist's attempt to keep his word to his friend, the late musician Gram Parsons, by burning his corpse in Joshua Tree National Park. Parsons' father wants a more conventional funeral, hence the need for the titular felony. Very loosely based on a true story; the movie adds a bitch-on-wheels ex as the main antagonist, presumably so the real (step)father who claimed the corpse (who probably has real lawyers) didn't have to be the heavy.
V gets one in V for Vendetta, his body is laid to rest in the train that delivers his bomb to the British Parliament.
Unsurprisingly considering it's a tale of an Arab experiencing Viking culture, two of these feature in the film version of The 13th Warrior, one near the beginning and one near the end of the film.

Literature

The probable Trope Maker is the funeral of King Scyld of Denmark in Beowulf. Scyld's funeral boat is not set on fire, though. Note that Beowulf himself is buried more conventionally, by cremation on a pyre and the remains being put into a mound.
A very fine how-to-guide appears in the 13th century Heimskringla in the description of the funeral of King Haki of Sweden. This variant involves fire.
In both the book and film of The Fellowship of the Ring, Boromir is laid to rest in one of the Fellowship's elven boats with both his sword, and the swords of his slain enemies (which was alot) The boat is pushed into the Anduin and allowed to tumble over the waterfall. It is not set on fire though. Given that J. R. R. Tolkien studied Beowulf extensively, it is hardly a mystery where he got the idea. Especially when we take into account his general love of medieval Norse culture.
The Silver Horde in Interesting Times plan to give Teach such a funeral. In fact they plan to give him every barbarian funeral they can think of.
Cohen: In a longship set on fire, on top of a heap of the bodies of his enemies, under a burial mound.
In the Doctor Who Expanded Universe short story collection Short Trips: Time Signature, several of the stories involve the Sixth Doctor trying to show his companion, William, some Vikings. He never succeeds, but after William's death, the Eighth Doctor takes his ashes to Vinland, and arranges for him to have a Viking funeral.
In the finale of the Conan story, "Queen of the Black Coast," Conan sees off one of his greatest loves, the pirate queen Belit, like this, turning her ship into a funeral pyre and sending it out to sea.
In A Song of Ice and Fire, this is the customary burial rite for the riverlands lords. Played for some laughs at Hoster Tully's funeral, when Edmure repeatedly fails to hit his father's boat with a flaming arrow.
In The Darkis Rising: Will gets the sixth and last sign from a great ship, carrying a long dead king who was a an ally of the Light (but not King Arthur), and all his possessions. After he claims it, in an act of spite the Dark sets the whole ship on fire. Will is horrified by it, but Merry points out that Dark was so eager to be spiteful that they didn't think it through. All they have done is give the King a Viking Funeral, which is exactly what he deserves.

Live-Action TV

In the Doctor Who episode "The Impossible Astronaut", the Doctor gets one of these. "Last of the Time Lords" featured The Master getting one as well.
In the Fringe episode "The Day We Died", Peter Bishop performs one for his wife, Olivia Dunham.
In Merlin, Merlin gives on to Freya, sending her floating on a lake before setting her body on fire magically, from afar.
Mysterious Ways: Performed for a character in "Something Fishy," although they light the boat on fire with torches before setting it out to drift rather than doing the flaming arrow thing.
Peggy: It's a Viking funeral.
Miranda: Yep.
Peggy: Isn't this illegal?
Miranda: Yep.
Spoofed in one episode of Malcolm in the Middle, when Francis discovers a dead janitor in a hidden closet at Marlin Academy. He decides, along with several other cadets, to give the corpse a send-off worthy of a Viking prince by loading it into a dinghy, along with the janitor's worldly goods, dousing the whole thing with half a can of gasoline, lighting it and giving it a good push into the lake. Unfortunately one of the other cadets has left the rest of the can of gasoline in the dinghy, and the whole thing drifts into Commandant Spangler's boathouse just as the can ignites, thereby causing it to explode.
During Season 1 of The Tribe, Zoot, the psychotic leader of tribe Locust, gets a Viking-style sendoff as well. This is mainly because the local cemetary has been overrun by members of a hostile rival tribe, but also because (as his brother Bray explains) " Zoot thought of himself as a warrior chief."
Played for laughs in one of Rose's stories on The Golden Girls. She says one of her relatives was being given a Viking funeral, but the fire department kept putting it out.
In one MacGyver episode, an evil therapist convinces one of his client to give himself a Viking funeral... And he takes a friend of Mac's to play the part of the slave.

Music

The Manowar ballad "Swords In The Wind" references the funeral-pyre version of this trope, owing to the band's general love of Vikings and warriors in general.
Place my body on a ship, and burn it in the sea
Let my spirit rise, Valkyries carry me
Take me to Valhalla, where my brothers wait for me
Fires burn into the sky
My spirit will never die!

Tabletop Games

Ruathym in Forgotten Realms apparently does it. At least, in a novel they solemnly cremated a captain with his ship. Unsurprising for the motherland of local Viking stand-ins.

Web Comics

Seen here in Homestuck. For a safe.

Web Original

Things Mr. Welch Is No Longer Allowed to Do in an RPG:
1018. Elves do not get Viking Funerals.
In the Strong Bad Email from Homestar Runner where the Paper is dying, Strong Bad actually responds to the Paper's death by burning it with his BMW lighter.

Western Animation

In an episode of The Simpsons, when Bart decided he needed to grow up and put away his childish things, he did so by giving them a Viking Funeral.
Justice League launched an actual Viking longship into the Sun.
- While Dianna read The Rime of the Ancient Mariner, for some reason. It does have very roughly the right portentous, marine feel, and the person in question had spent centuries trying to die, so points for the vague Flying Dutchman tie-in, but viewers who recognized the poem were still somewhat taken aback. What kind of elegy is that for a Viking prince?

Video Games

The Lost Vikings being, well, Vikings, send their fallen off on a burning ship on the game over screen.
It appears to be tradition in Ferelden. After the siege, the dead of Redcliffe are sent off in floating pyres. In the Return to Ostagar DLC, this is the most respectful send-off you can give the late King.
Ezio gives one to his father and brothers in Assassins Creed Brotherhood.
The video game based on Zemeckis' Beowulf, like the movie, features one in a cutscene. Different from the film, the ship is kindled by pouring flaming oil onto it as it passes beneath an arch.

Villain's Dying Grace