Time Travelers Are Spies

... Or at least they're likely to be mistaken for them whenever the era they travel to is familiar with the concept.

Think about it. If you're from a different time, then you:
 * Possess a bunch of knowledge that most people wouldn't have, some of which may very well be classified in the time you've arrived in.
 * Lack a bunch of knowledge about pop culture and current events that most people would have.
 * Probably have an accent that sounds slightly off.
 * Can't convincingly account for your past, because you literally don't have one in that timeline.
 * Have goals which probably look incomprehensible to other people, and therefore are likely to be taken as cover for something more sinister.

All of these will make you appear to be a spy to anyone who takes enough of an interest in you to notice.

Compare Fish Out of Temporal Water, the root cause of this trope.

Comic Books

 * This happens to Jimmy Olsen in a Silver Age story when he gets transported back to World War Two and has to pose as a Nazi.
 * The EC Comics story "...For Us the Living" (Weird Fantasy #20) begins with an atomic scientist being arrested as a spy for a foreign power. The scientist admits his identifying documents are all forged because he came from an alternate time-branch in which Abraham Lincoln escaped assassination and brought peace to the world.
 * Mortimer experiences this twice in the Blake and Mortimer album The Time Trap.
 * In an Archie Comics comic, Jughead accidentally travels back in time to the American Civil War and is mistaken for a Confederate spy (one of the suggestions being that the S on his shirt stands for "Spy" or "South").

Films -- Live Action

 * In Hot Tub Time Machine, Blaine and the ski patrol mistake the main characters for Soviet spies after finding their cell phones and MP3 players (which they think are spy gadgets), and their can of Chernobyl energy drink with its Cyrillic lettering (which they think is a bomb).
 * Chekov and Uhura in Star Trek IV: The Voyage Home, in the famous "Nu-cle-ar wessels" sequence. Somehow they don't realize that, if you want to get aboard an aircraft carrier in 1986, the guy with the prominent Russian accent probably shouldn't be your spokesperson. This doesn't cause them any problems in the end except for a bunch of strange looks, but when the inevitable transporter malfunction strands Chekov next to the reactor of aforementioned nuclear wessel he is assumed to be a Russian spy. His recitation of Name, (Starfleet) Rank, and Number do nothing to dispel this.

Literature

 * In Johnny and The Bomb, both Bigmac and Wobbler are mistaken for spies when the main characters travel back in time to the Second World War. As in some other examples, the WW 2 people regard their modern technology as spy gadgets, especially a pocket radio. It doesn't help that it says "Made in Japan".
 * A frequent plot device in Harry Turtledove's Crosstime Traffic young-adult series. While not exactly time travel, the protagonists travel to alternate universes where historical events have departed from their own Future History. In most of the stories so far, they get mistaken for spies. For example:
 * In The Disunited States of America the protagonist
 * In Curious Notions the protagonists
 * In The Gladiator the Crosstime Traffic organization.
 * The end of Robert A. Heinlein's Time Enough for Love involves the protagonist travelling back to the time of his own childhood and dealing with the difficulties thereby. Despite being a 2000+ year old pansexual incestual time traveller speaking what, after 2000+ years, might as well be a foreign language in a very alien culture, he encounters no problems. However, his plan to escape having to fight in WWI is to hustle pool and flee to Brazil. He then realizes that that might get him killed by German or French agents as a spy.
 * The Time Scout series doesn't say you'll be caught as a spy, but the results are the same. Get caught, get killed. Usually for apostasy or some such.

Live Action TV

 * In the Eureka episode "Founder's Day", a bunch of characters from the present day appear at an army camp in 1947; the general in charge of the camp spends most of the episode chasing them around assuming they're spies.
 * They are. They are in disguise trying to obtain information. They are just not hostile spies.
 * In Stargate SG-1, the team finds themselves at the bottom of a missile silo in 1969, leading naturally to this assumption. It doesn't help that the officer who just spotted them asks them, in Russian, "Are you Russian spies?" and Daniel promptly replies, "Nyet." O'Neill is not happy.
 * This happens to Gary in an early episode of Goodnight Sweetheart. He decides to play it up and manages to convince his captors he is a spy, but a British one and not a German one as they'd assumed.
 * This is the premise of the K9 episode "The Cambridge Spy": a lightning strike sends Jorije back to Cambridge in 1963, she's arrested as a Russian spy, and K9 and Starkey have to travel back to rescue her.
 * Star Trek: The Original Series:
 * "Tomorrow Is Yesterday". When the Enterprise accidentally travels back in time to Earth in 1969, Captain Kirk is considered a spy when he's caught infiltrating a U.S. Air Force base. (When an interrogator threatens to lock him up for two hundred years, Kirk ruefully acknowledges, "That ought to be just about right.")
 * "Assignment: Earth". When the Enterprise is deliberately sent back in time to Earth in 1968, Kirk and Spock are arrested as spies when they're caught inside McKinley Rocket Base.
 * The Doctor and his companions in Doctor Who sometimes fall prey to this, such as when they go to Skaro in "Genesis of the Daleks," where they are mistaken for the Kaleds' deadly enemy the Thals (or their common enemies, the wild mutants that run around in the nuclear wasteland). Everyone being Human Aliens in this case did not help. Sometimes the Doctor genuinely is a spy, working for himself, simply by virtue of being too nosy for the villain-of-the-week. Other times, such as in "The Brain of Morbius," the irate locals correctly identify him as a Time Lord but automatically assume he's there to steal their stuff because they're suspicious of Time Lords in general. The titular Morbius similarly assumes the Doc is there on behalf of the Time Lords to hunt him down and was just stumbling into the creepy castle as a ruse.
 * Played for Laughs in an episode of Boy Meets World, in which an accident involving a microwave oven sends Cory back in time to a warped version of The Fifties. (It's All Just a Dream.)
 * Happens multiple times in The Time Tunnel.

Tabletop Games

 * In the GURPS Infinite Worlds setting, this trope is played with due to the fact that Infinity Patrol agents routinely do become spies in the Alternate timelines they infiltrate, both to avoid the notice of the locals, as well as to hide from agents of rival Crossworld travelers from the Centrum timeline.

Video Games

 * In Back to The Future: The Game, Edna Strickland is convinced that Marty is a spy named Yakov Smirnoff (partially because a different time traveller told her this.) Additionally, Arthur McFly is convinced that Marty is an FBI agent.

Webcomics

 * Ophelia (a modern-day office worker) gets mistaken for a spy in this strip of Irregular Webcomic, because she knows that the Nazis never planned to steal the British Crown Jewels.

Real Life

 * The claims made by John Titor fall into this category. He might be just another Usenet crank, a very skilled practical joker, or the real thing.
 * Recently, a mysterious figure in a Charlie Chaplin film raised eyebrows.