Wow, they make you completely forget about the giant hawk wings, don't they?
"Your beauty is beyond compare With flamin' locks of auburn hair With ivory skin and eyes of emerald green" — Dolly Parton, "Jolene"
Green-eyed redheads are among the rarest color combinations found in humanity (with the exception of some abnormalities), and thus is used to distinguish an important, notable character visually. It's more often found with women than with men, and is almost always either found as a trait of The Hero or The Hero's Love Interest. To qualify for this trope, it's important that:
The character isn't from a live-action work, unless they were artificially given these colors for the purpose of the work, or were chosen directly for it.
The character has red hair and green eyes, and not pink hair or hazel/blue-green eyes, or even green hair and red eyes.
This image might have been influenced a lot by Maureen O'Hara and her various roles, which also influenced this to be the Phenotype Stereotype for an Irish person. It may also have something to do with the fact that red and green are complementary colours - which also may help to explain why so many red-headed characters are depicted wearing green, even if they aren't Irish.
Examples
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Anime & Manga
Kurama, at least in his human form. The red hair, green eyes combination in this case is a warning not to underestimatehim.
Hellsing's Pip Bernadotte, Love Interest of Seras Victoria. While he's a mercenary and an Anti-Hero to be traditionally sweet, he is the nicest of the guys in these series along with Sir Shelby Penwood. With the other men being Dracula, a deadly butler, Nazis and Fanatic Vatican Crusaders, Pip looks like a saint.
Kino Makoto/Sailor Jupiter from Sailor Moon is this in the manga, Jupiter moreso. Word Of God says that her eyes turn green after transforming into Jupiter, and they look more hazel/brown in her untransformed state in some artwork.
Lyra, daughter of Thundra and The Hulk, combines this with green skin.
Lena Luthor, at least in Supergirl: Cosmic Adventurs in the 8th Grade
Also the "Lex's daughter" incarnation in The DCU ... although in this case her green eyes may have indicated that she was still connected to Brainiac.
Lex Luthor, usually acknowledged as Superman's arch-rival, has naturally red hair (though he's usually bald for a number of reasons Depending on the Writer). Incidentally, so does Superman's best friend, Jimmy Olsen.
Caitlin Fairchild from Gen13 is this physically as well as usually personality wise (though she can be a Fiery Redhead when it comes to protecting the others).
Poison Ivy.
Rogue's hair is usually referred to as "auburn", but was colored more red in earlier comics. And her eyes are green.
Starfire, with golden skin.
Leetah and her daughter Ember, as well as Redlance and his daughter Tyleet in ElfQuest. Notable because Leetah and Ember are both dark-skinned.
Merlin in the 1998 miniseries, where it appears that both young and old Merlins' eyes were digitally enhanced to be green beyond reality.
Literature
Mara Jade from the Star Wars Expanded Universe may be the ultimate example: a Badass and former Evil Redhead who was so enormously popular, she joined the good guys, became The Hero Luke's long-time partner, and was eventually happily married to him. Until Legacy Of The Force, but no one likes what happened there.
Scout, the Jedi Padawan equivalent of Batman in Yoda: Dark Rendezvous, has these.
Becky Sharp in Thackeray's Vanity Fair (1847-48) is a green-eyed Evil Redhead, though her hair is probably more of the sandy variety.
Beatrice Bailey, the protagonist of The Witches Of Bailiwick. Her specialty is controlling the weather, which goes with the nature aspect of this trope.
Christopher Hitchens mentioned this trope while criticizing a stereotypically Irish-American character in John Updike's last novel.
The title character of Fire is red-headed and green-eyed and beautiful beyond the lot of ordinary people; her hair and eye color both coming from her mother.
Cherry Valance from The Outsiders is the redhead with green eyes that starts all the trouble in the book, while allowing Ponyboy, the protagonist, to change his view on Socs.
In Michael Flynn's January Dancer, a woman is introduced by her red hair and green eyes even before her name is given.
The Girl from The Club Dumas; her very light green eyes are mentioned many times as being captivating, likely because she is in fact a fallen angel. In the film adaptation, The Ninth Gate, however, she has different coloring.
Mildmay, one of the viewpoint characters of The Doctrine of Labyrinths. For most of the story, he's one of only two-redhaired people, the other being his brother, and the only redhead with green eyes. Felix often describes them as being like jade.
Live-Action TV
Doctor Who has Amy Pond, whose actor has actual red hair and green eyes.
Music
In Dolly Parton's song "Jolene", the titular character (and the narrator's romantic rival) has "flaming locks of auburn hair" and "eyes of emerald green".
William Sheller's song "Les miroirs dans la boue" (The Mirrors in the Mud) is all about a young girl with "green eyes bathed in red hair" ("les yeux verts noyés de cheveux roux").
Gaelic Storm's Green Eyes, Red Hair is all about this. The woman in question overlaps with Fiery Redhead too.
Mythology
Thor in Norse Mythology is a rough tough badass god, who happens to be redheaded and green-eyed. Yet when he had to cross-dress as a woman in Thrymskvida, all the giants were so smitten by "her" dazzling looks that they failed to notice many, many less-than-subtle hints of a trap...
Video Games
Talia Maurva from the Aveyond Series has red hair and green eyes, and is a talented magic wielder.
The Bridge Bunny Kelly Chambers in Mass Effect 2 is probably the most trouble-free and relaxing romantic interest (for both genders). Don't expect any ass-kicking from her, though.
Default female Shepard is another example. Expect muchasskicking from her.
Aveline Vallen in Dragon Age II, is the first companion you pick up and is a green-eyed redhead.
If you squint hard enough, you'll notice that pre-zergification Sarah Kerrigan from Starcraft series had grayish-green eyes (especially on that photo Raynor has of her in SC2), and she was a redhead.
Luke Fon Fabre from Tales of the Abyss. This is actually a plot point, as a noble with red hair and green eyes has a great role in a prophesy that will lead to world prosperity.
Also in DCAU, Pamela Isley, a.k.a. Poison Ivy, in the Batman episode "Pretty Poison", when she uses her stunning looks to get close to the men who wronged her (one of them even proposes to her) and attempts to assassinate them.
Gargoyles has multiple examples. Demonain her human form is one of these, and so is David Xanatos' wifeFox. Also Macbeth's wife Gruoch.
