In Iron Man, Tony Stark shaves off his beard and dyes his hair blond while on the run. It works, as no one recognizes him, but he feels he looks like "an idiot".
In Omaha the Cat Dancer, the title character bleaches her hair blond before fleeing Mipple City after she and Chuck had a severe falling out. Eventually, she decides to return and goes to her favorite hair stylist to put her hair back to normal. That proves a very unpleasant experience even as the stylist makes her opinion clear that Omaha shouldn't have abused her hair like she did, in other words, "Next time you get an idea about your hair? Call for a second opinion, Okay?"
Done in Watchmen by The Silk Spectre II and Nite Owl II, when they assume new identities at the end, having allowed everyone to assume they died in the Big Bad's attack on New York. They both dye their hair blond. Also, Nite Owl II grows a mustache and Silk Spectre II cuts her hair.
Played for laughs in Why I Hate Saturn: brunettes Anne and Laura are on the run from Laura's ex-boyfriend and they dye their hair blonde so they won't be recognized. When he finds them anyway, they run again, and Laura decides to return her hair to its original color - only she messes up and it comes out green!
Green Arrow shaved his hair and beard during Mike Grell's Black Arrow arc.
Done by Evelyn Salt in the movie Salt. She dyes her blonde hair brown. And at the end disguises herself as a man to sneak into the White House.
Agent 99 in the 2008 Get Smart movie - but her dye job is incidental next to the extensive plastic surgery she undergoes to keep her job as a secret agent.
In Dutch WW2 drama Zwartboek AKA Black Book in order to spy on the Nazis, the Jewish main character Racheal had to dye her brown hair blond, and not just the hair on her head (since this is a Paul Verhoeven film, guess which part was shown).
In Species, after faking her death in a car crash with one of her own severed thumbs Sil cuts her long blonde hair shorter and dyes it black to hide from the scientists who are hunting her.
Inverted in The Long Kiss Goodnight where the lead's usual hair is dyed blonde during her time as an assassin. When she loses her memory, she lets her natural auburn hair grow out. The CIA is still able to recognize her on video, despite a drastically different style of clothing as well.
Attempted by a cavewoman mother in When Dinosaurs Ruled the Earth as that tribe had a tradition of sacrificing blondes to the sun. Of course, she attempted said dye job with tar.
Emilie in Lockout. She ends up the only woman in the midst of a prison riot. Snow does said dye job with coffee, motor oil and other nasty stuff. Oh and a punch to the face to make her look more masculine.
In Elektra Abby is told to do this while on the run from The Hand. She dyes her hair brown and cuts a fringe to resemble Elektra. Oddly enough Mark and Elektra don't do this - and they're still found pretty soon too.
In Following Bill gets paranoid about being recognised by someone whose house he robbed. Cobb then convinces him to shave, cut his hair short and don a different wardrobe. Of course this bites Bill in the ass when Cobb frames him for everything.
Played with in Shadowboxer. Vickie's hair is cut while she's on the run but she wears a blonde wig instead of dyeing. And after a few years have passed, she stops wearing the wig and her hair has grown back.
In Breakout Jocelyn cuts her blonde hair to shoulder-length and darkens it to brown after she has fled from her abusive husband.
Julia Roberts's character in Sleeping With The Enemy plays with this. Although she cuts her hair after faking her death, it's presented more as an Important Haircut and she doesn't look that much different afterwards. She wears a wig when travelling to her new town too.
The Lifetime movie Black & Blue. Gracie flees her abusive husband and is advised to cut her hair - because it's too conspicuous.
In The Outsiders, Johnny cuts and dyes Ponyboy's hair blonde when they go on the run after Bob's death.
Sansa has the distinctive Auburn hair of the Tullys. So she dyes her hair a more normal mousy brown while disguised as 'Alayne Stone'.
Princess Myrcella has her distinctive Lannister hair dyed and her companion stands in for her while travelling to Dorne.
'Young Griff' dyes his hair blue in memory of his Tyroshi mother. Or so he claims. It's later revealed to be in order to disguise the distinctive silver hair of (it's claimed) Aegon Targaryen.
An earlier Aegon Targaryen cut his off instead.
Abby in The Firm.
Zakath grows a beard as part of not being recognized in The Malloreon. Belgarath dyes Garion's sandy blond hair dark brown for the trek across Morindimland. Polgara has tried dying her hair in the past to hide, but her Skunk Stripe always comes back and makes it impossible.
Reversed nicely at one point in the prequel novels. Dye won't adhere to Polgara's white lock, and she's trying to be undercover. So Belgarath (in his wandering storyteller disguise) starts telling ladies across the western kingdoms that the latest fashion in hairdressing is dark hair with a dyed-white streak in it.
In The Awakening by Kelley Armstrong, Chloe's blonde-with-red-streaks hair is recognized by a by-stander, and so she is forced to dye it black. Since she's a sheltered fifteen-year-old and the new color looks atrocious, this is very upsetting to her.
In the Holocaust-set novel Number the Stars, Peter has distinctive red hair and must keep a hat on it to keep the Nazis from finding him.
Somewhat like the Polgara example above, it's mentioned a time or two in the Heralds of Valdemar stories that Companions can't be dyed — or rather, the dye job never lasts long enough to do much good.
In I, Jedi Corran Horn dyes his hair and grows a goatee in order to attend Luke Skywalker's Jedi Academy without openly being a hero of the Rebellion and drawing too much attention. The dye, being special dye with a metabolizing agent, is initially misused, making him green-haired from head to toe, and he has to call his old Space Cop partner for help.
The cyborg disguise is a Call Back to Wedge's Gamble where he impersonates the same person when infiltrating Coruscant.
Used by Ani (Isi) in Shannon Hale's The Goose Girl. Since only her countrymen (read: the traitors that want to KILL her) have blond hair, she tries to dye her butt-length yellow locks. Subverted in that she can't find enough brown dye, as it's uncommon in Bayern, and settles for wearing a scarf and only dying her eyebrows.
Both Aiden and Meg Falconer in On the Run. Meg bleaches and cuts her dark hair, while Aiden dyes his blond hair brown.
In Dragon And Phoenix Maurynna has to dye her hair and skin to traverse part of Jehangalan without drawing attention to the fact that her ethnicity isn't represented there at all. Her friend Raven wants to do the same, but the dyes would not work well on his brilliantly red hair or his pale, freckled skin.
In The Guns of Navarone, Dusty Miller and Andy Stevens both have their hair dyed black so that they can pass for Greek fishermen, instead of the saboteurs they actually are. After the scene on the cliff, Captain Mallory notices that the rain is starting to wash the dye out of Stevens' hair, and is furious because their superiors supplied them with low-quality dye, which could cost a man his life if they were discovered. This was somewhat of a moot point, since Stevens was already half-dead from falling into a ravine, and they could no longer have passed for fishermen at this stage in the plan anyway.
I Am Number Four - Number 4/John Smith bleaches his dark hair blond when he has to move cities (ironically inverted in the film where he is played by the naturally blond Alex Pettyfer). He has to do it again in the next book Power Of Six when he actually is on the run from the law. Him and Sam are forced to shave their heads. Number Six doesn't have to do this since she has the power of invisibility.
Rose Madder - Rose isn't trying to disguise herself when she impulsively dyes her brown hair to blonde, but the dye job does save her at least once from being recognized by her Ax-Crazy husband when he's hunting her in order to kill her for leaving him.
Elizabeth does this in Tell No One after faking her death. Also owns a collection of wigs just to be safe. Her counterpart in the French film adaptation doesn't do this.
Live Action TV
Used in Heroes, season one, 'Five Years Gone.' Claire's iconic blond locks have been dyed brown.
Done in the TV adaptation of the Covert-One novel The Hades Factor, with the character in question (going from redhead to blonde) referred to as the "Killer Blonde version".
Doctor Who: In "The End of Time", the Master is assumed to have dyed his hair to avoid being recognised as the former Prime Minister. In reality it turned white because his resurrection was screwed up.
In Hyakujuu Sentai Gaoranger, Gaku Washio (GaoYellow) bleaches his hair to disguise himself because he's officially AWOL from the Japanese Air Force
In the Highlander episode 'Methuselah's Gift', Amanda does this when she's being chased by mortals who want her crystal. She bleaches her dark hair white.
Attempted by a suspect in NCIS. She walks right past Tony and Ziva thanks to a dye job, but they catch her after they find hair dye residue in the hotel room she had recently vacated.
Judging from previews for the eighth season premiere of Bones, when Brennan became a fugitive after being framed for murder, she used peroxide on her hair to pass for blond as a disguise.
Heather/ Cheryl/Alessa, in Silent Hill 3, has her hair dyed blond as part of her disguise to avoid the titular town's cult. Her father probably does this from the time she's young, and she apparently continues doing it when she gets older.
She is a teenager, however, so it might have been a style choice.
Sonny Bonds needs to dye his hair as part of his undercover disguise in Police Quest.
Max Payne shaves his head as a quickie disguise in Max Payne 3. He admits that it's kind of a stupid disguise, and he gets made fun of for it later.