"Here's how great it is to be white — I can get into a time machine and go to any time and it would be fuckin' awesome when I get there! That is exclusively a white privilege! Black people can't fuck with time machines. A Black guy in a time machine is like, 'Hey anything before 1980, no thank you, I don't wanna go.'"

—Louis C.K.

Modern Western society is ostensibly an equal opportunity environment; most fiction writers suggest that the sci-fi future will be even more so. But if time travel ever becomes an institution in the future, some parts of the past may not be safe for all people to travel to.

Imagine being a black man and traveling to a place and time when all blacks were assumed to be slaves and had to carry papers to prove otherwise, or where they were likely to be lynched for speaking up. Similar issues exist for other races too — a white guy (or Japanese guy) in China during the Boxer Rebellion is sure to attract unwanted notice. Similarly, being a Semite, Roma, or homosexual and traveling to Nazi Germany (or any Nazi-occupied area, for that matter) is a really bad idea, unless you want to experience a very, very painful death. If they go to these dangerous time periods anyway, expect repercussions. Women in many eras and places will have similar issues, although to a lesser degree — the culture shock of a less liberal society may be a plot point.

How realistic (at least, to the extent that a story about time travel can be rooted in realism) this trope is varies. There is no use pretending that racism did not exist, but the levels and expressions thereof have varied wildly throughout history - it is not cleanly divided the dangerous Past and the accepting Present.

Note that one should be careful not to generalize, as bigotry was never universal even in eras when it was at its height. Hillsdale College, a liberal-arts school in Michigan, was admitting women, Jews, atheists, and people of color as students before the Civil War.

Compare Black Vikings, Politically Correct History. Also Eternal Sexual Freedom.

Examples:

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Comic Books

One arc of JSA saw some of the team sent into the 50s to the time of the original Justice Society. The black Mr. Terrific had some unpleasant experiences in the pre-Civil Rights era, like being forced to change train cars, but took it rather stoically. And then, just to rub it in, he fights a KKK chapter who manages to get a noose around his throat.
In Yoko Tsuno last story involving time travel, Monya points out that it's easier to walk unnoticed in medieval China without Yoko's European friends. A little odd, because usually the whole gang traveled, but now they have so many extra members that there is a sufficient team without them.

Film

Timeline: An intra-Caucasian example: In the movie adaption of Michael Crichton's novel, one of the time travellers gets killed by 14th century Englishmen (who are at war with France), after they forced him to confess his French nationality, which they already did guess based on his accent. *
Which doesn't make any sense, since in the 14th century, the English ruling class was itself still French-speaking. It was only over the course of the Hundred Years' War that the English ruling class became Anglicized.
Variation in Escape From The Planet Of The Apes Zira and Corenlius are talking apes from the future, where their kind rule the planet. When they travel back in time, they're in 1970s USA, where humans rule and apes are wild animals, resulting in them being taken to a zoo and assumed to be animals until they reveal their secret.

Literature

Octavia Butler covers the perils of time traveling while black in the 1979 novel Kindred - the black protagonist goes to 19th-century Maryland to meet her ancestors, one of whom is a white slave owner. Drama ensues.
In To Say Nothing Of The Dog by Connie Willis, the time-travel research division at Oxford contains a black student and a South Indian professor, who can't do much actual time-traveling (all of the work at Oxford seems to do with England, for some reason) for safety reasons. It works to their advantage, as the plot of the book involves everyone else in the department being forced to do far too much time travel for their own health.
In an Animorphs story involving time travel, the group runs into this problem — they wind up at Princeton University, circa 1934, in an alternate timeline with a lot of differences from the real one- where someone calls Cassie something she would REALLY rather not be called. She puts him in his place to the tune of a 900-pound Polar Bear.
Cassie: "You don't like black people, Mr. Davis? No problem. I can turn white. Watch me."
- Random trivia: Though they have white fur, polar bears in fact have black skin.
  - To Mr. Davis' credit, if I had just called someone THAT and then saw her transform into a 900 pound (very angry at this point in the story, I might add) Polar Bear, pointing out that technically her skin is still black would probably rank pretty high on the "Shit I will never say if this happened to me" list.
In Johnny and the Bomb, Yo-less travels back a mere forty years or so, to World War II era, and has to deal with people calling him "Sambo".
Inverted in Andre Norton's Time Traders series, where both the titular organization's agents and their Soviet counterparts are sent on undercover missions in different areas and eras of history (and their cover identities composed) specifically on the basis of their racial makeup. Not only is conspicuous behavior avoided for fear of the usual Butterfly of Doom, but because word of it in history books would alert the enemy to your position in time.
The book A Wish After Midnight sends a girl back to the Civil War.
Pretty much the entire point of Harry Harrison's A Rebel In Time. White supremacist goes back in time to help the South win the Civil War, and a black FBI agent decides it will be a snap to follow the killer back to antebellum deep south. When the slave owners of the era see his high quality clothes and hear his 20th century New York college educated accent, things proceed pretty much as you'd expect under the circumstances.
In Orson Scott Card's Pastwatch: The Redemption of Christopher Columbus, only one of the three time travellers sent to change the result of Columbus' first encounter with native Americans is the appropriate race for the culture that they will be dealing with (and none of them are white). In the end, one character, a black woman, has to overcome a lot of prejudice to win over the natives, but she eventually succeeds while another, a middle-eastern man sent to sabotage one of Columbus' ships, actually uses his race to the mission's advantage, revealing himself and allowing his Heroic Sacrifice to unite the crew against the "Muslim enemy" Even the Mayan going back to visit the Mayans is a foot taller than the Mayans of that time, making him stand out.
Tunnels Of Treachery plays with this - the two previous books both involved white kids going back in time using the Moose Jaw tunnels, and when their Chinese-Canadian friends do so, they get a very different reception.
Averted in S.M. Stirling's Nantucket series: Capt. Alston, an African-American Coast Guard officer, is assumed, by the Bronze Age people she encounters, to be a respected Nubian warrior chief.
In Time Scout, Women cannot be scouts. Period. Until Margo insists. They can be guides. Guiding and scouting are wildly different professions. The race issue is never brought up.

Live Action TV

Martha Jones on Doctor Who tends to get away with this for the most part, though her trips are rarely to the distant past and when it is, the issue will be addressed. "The Shakespeare Code" had Martha worried about being sold as a slave, but the Doctor assured her this wasn't actually an issue. In reality, there actually were some black people in England, none of whom were slaves, and the dialog was actually meant to teach kids that England wasn't entirely white in the 17th century. Furthermore, the Doctor apparently covered Martha's background as Shakespeare later asked her tell her about a country where women could be doctors. The later "Family of Blood" two-parter had Martha's race subtly addressed as nobody believed a woman, let alone a poor woman (Martha had to pose as a maid), let alone a minority was capable of being a doctor. Martha got a Crowning Moment of Awesome proving her extensive medical knowledge to one such doubter.
- Donna, while not a minority, has some issues in ancient Pompeii due to her blunt personality.
- For a series of the Big Finish Eighth Doctor line, the Doctor's companion C'rizz is a Rubber Forehead Alien whose skin changes color like a chameleon. The most attention this attracts is when he's lured into becoming a sideshow attraction in a Victorian-era circus.
- The early Hartnell series sometimes avoided this by having the female characters dress as men. This happens in the Crusade and the Smugglers. The Massacre has no female companion in the main story.
- Also in the Doctor Who spinoff The Sarah Jane Adventures episode "The Temptation of Sarah Jane Smith," when Rani goes back to the 1950's looking for Sarah Jane.
  Rani: "Yes, I get it, ethnic person in the 50's!"
  - Although this is a subversion, as after she leaves, the punchline is 'What on earth was she wearing? Can that really be the fashion in the Punjab?', implying that not only were they staring at her clothes, not her, but her ethnicity is the only reason she wasn't called on her outfit, as they assumed she was from India. While 1950 English people might be somewhat prejudiced, they would hardly be startled by someone of Indian descent simply existing in England, but would be utterly flummoxed by a girl dressing like that.
- Some of the Eighth Doctor Adventures novels have this become a bit of an issue for Anji Kapoor. In the 18th century and again in the 19th century, she's treated as exotic and mystical but not outright abused. Wearing a sari helps; too bad she hates wearing saris. Fitz Kreiner, who's white and British, gets almost as much trouble for stuff like his lower-class London accent.
Amanda's black roommate in Lost In Austen points out that she can't go through the door because she's black.
An third season episode of seaQuest DSV, the titular sub ends up in the 60s during the Cuban Missile Crisis. Commander Jonathan Ford (a black man) takes a team to the surface. They "borrow" a car and take it to their destination on the shore. On the way, they pass by car full of young men. They arrive to the beach only to see the other car pull up behind them and the guys getting out with baseball bats. Ford suddenly notices a "No blacks allowed" sign and remembers his history. Plus, he was in the same car as a white woman, which only pissed off the 60s guys more. Luckily, all of the team members are military-trained, so a bunch of punks with baseball bats is not a threat.
Subverted (Inverted?) in the Stargate SG-1 episode 1969 The hippy calls Teal’c “brother” and insists that ride up front with him.
- Averted but discussed. The hippy is making a point of showing that he's not racist, unlike a lot of his contempories.
Subverted in a Chappelle's Show sketch, where "Playa Haters" go back in time and shoot a Southern plantation owner.
Done with space travel rather than time travel on the original The Tomorrow People, when the characters visit a planet of Human Aliens. As there are no dark-skinned people on that world (or at least that part of it), a black character from Earth isn't able to accompany her companions in public.
Averted on the Next Gen episode "Time's Arrow," where the android Data is sent back in time to late-1800's San Francisco. His Starfleet uniform gets more attention than his albino-pale skin and yellow eyes, and he's able to pass without trouble by telling everyone he is from France. (It helps that, having a perfect memory, he can speak perfect French.)
This trope was the center of an early Quantum Leap episode, where Sam ends up in the body of a black man, in the past. Sam nearly gets himself into hot water immediately by trying to sit down in a cafe and order a meal.

Magazines

Contact magazine, based on the 3-2-1 Contact TV show, had a recurring segment involving two time-traveling teenagers. One installment had the (presumably white) American kids get in trouble when they ended up in Japan during World War II. In Hiroshima just before the bomb was dropped, no less.

Stand Up Comedy

Comedian Louis CK points this out when mentioning how great it is to be a white man. A white man can get into a time machine and go just about anywhere at any time and be welcome. If you're black, you wouldn't want to go anywhere earlier than the 1980s. On the other hand, he wouldn't want to go into the future, since Karma's coming to bite the white man... hard.
- Of course, this shows a drastically limited imagination, as there are a lot of times and places where a white man would not be welcome, and a lot where a black man would be just fine. The quote only makes sense if you're considering post-colonisation USA as the entirety of the world.
One black comedian made the same point about American nostalgia.
You know why black people don't understand the Republican party? It's simple; they're always talking about the 'good old days'. We didn't have any good old days! 'Hey, remember the back of the bus? You could always find a seat on the back of the bus.'

Video Games

World of Warcraft uses a Fantastic Racism variation of this trope: player races that were not a part of the Alliance prior to the Third War are given a race-lift while running certain instances in the Caverns of Time, so that they avoid attracting unwanted attention.

Western Animation

In one episode of X-Men, a few alternate-universe versions of the X-Men travel back to the 50's to save the younger Professor Xavier from a time-traveling assassin. They all talk at a cafe, and the owner gets pissy about the fact that Storm and Wolverine (an African and a Caucasian, respectively) are a couple. Naturally, this makes Wolverine completely flip out.
- What, didn't he notice that Wolverine radiates "Don't fuck with me" vibes?
- Not surprising given that the X-Men are one big metaphor for racism and prejudice. Though Storm is more amused than offended—after facing persecution her whole life for being a mutant, she remarks that plain old fashioned racism is almost quaint.
- Amusingly also something of an inversion, as this also gets them mistaken for beatniks, resulting in a patron with beatnik sympathies siding with them in the ensuing brawl
Family Guy episode "Road To Germany". Mort, who is Jewish, accidentally activates Stewie's time machine and is set to Poland on the eve of the Nazi invasion. Stewie and Brian go back in time to rescue him. At one point they need to pass as Nazis themselves, and having Mort the walking stereotype in tow proves problematic. At one point, they tried to pass Mort off as a Catholic priest. And then he's asked to give someone their Last Rites.