Anachronism Stew is common in the works of William Shakespeare, because theatre of that time took a completely different approach to historical drama. Regardless of when or where a play was set, costumes and patterns of speech from the (then) present day were used, and there was never any attempt at historic realism as we understand it today. Some examples:
Julius Caesar, which contains references to striking clocks despite the fact that the first mechanical clock would not be invented until the mid-13th century.
Julius Caesar makes reference to a doublet, a close fitting jacket that wasn't around in Roman times.
There is also the coffin in which Caesar's body is placed. Ancient Romans never used coffins at any point in their burial practices.
This is the result of massive Latinization or Hellenification of the past names, and obsessive attempts to match pantheons of other countries to the Greco-Roman one in the Middle Ages, result of the ideas of the superiority of the Latin language - scholars even made attempts to change English grammar to match Latin! The same process had been going on even back when the Roman Empire and Greek nations still thrived.
Titus Andronicus is filled with them. The play is set some time in pretty generic Ancient Rome and it is filled with references to Christianity.
Hamlet (fl. 12th century) is a member of a religious denomination that won't exist for 300 years and attends a university that won't be founded for 200 years...
Even worse: Hamlet is based on events taking place in 8th century Denmark, and Denmark didn't convert to Catholicism (the only game in town) until the middle of the 10th century. So the original Prince of Denmark was not Christian at all. Shakespeare made him Christian to make him more interesting to Shakespeare's audience.
If some scholars are to be trusted (Most prominently J. R. R. Tolkien), the exact time of Hamlet is the fourth century AD, him being the last prince of Jutland before the Danes came along...
There's an essay on Hamlet that points out that his religion is actually key to interpreting his actions. If he is a Protestant then he believes the virtuous dead go straight to heaven, and therefore his father's ghost must be damned (and, presumably, completely unreliable). If, on the other hand, he's a Catholic, then he'd expect his father's soul to be in purgatory, and therefore a credible witness to Claudius's misdeeds.
Many performances of Troilus and Cressida deliberately use this trope by placing the heroes of the Trojan War into settings like World War I style trench warfare, in order to emphasize parallels with modern war. The play itself has an interesting and subtle example - Hector covets the fancy armor of a Greek soldier, but the few descriptions of the armor indicate that it is clearly in a modern British style instead of ancient Greek armor.
Opera also does this with contemporary music played and sung in an earlier setting. Some examples include:
Richard Strauss's Der Rosenkavalier, which is set in late 18th Century Vienna, yet uses and have the characters sing to a Viennese waltz, which only became popular in the mid-to-late 19th Century.
Medieval mystery plays did this deliberately — either to emphasize relevance to contemporary concerns (King Herod was recast as a scheming aristocrat sending out his knights to kill babies to protect his power base. Also, he was a Muslim), or just for comic effect (Noah exclaims "By St. John!" while arguing with his wife; the shepherds invoke about 5 different saints, the cross of Christ and the Virgin Mary before the angel turns up to tell them that a saviour has been born in Bethlehem... which is within walking distance, despite the fact that the shepherds have mentioned that the action is taking place in the vicinity of the English village of Horbury).
Similarly, to this day actors in most Peking Operas are dressed in Ming Dynasty costumes no matter when the story is supposed to take place.
Probably played with a lot in Japanese Kabuki theater due to government restrictions on content, costumes, and hair styles: a play that referenced a current issue would claim to be set in another era, except the characters might just happen to be wearing contemporary clothes.
The musical adaptation of Spring Awakening is based around this trope. While taking place in a provincial German town in 1890, in moments of emotional intensity, the characters whip out microphones to deliver interior monologues in rock music fashion, complete with concert lighting. These songs make no attempt at being time period appropriate: the characters sing in modern slang and the lyrics mention telephones and stereos, among other things.
The 1971 Broadway adaptation of Two Gentlemen of Verona spliced early-seventies rock music into Shakespeare in a similar fashion.
One filmed version had a scene in which Judas was chased by helicopters and tanks.
Although it's set in 1587, the male (and one female) characters of Mary Stuart wear relatively modern outfits (e.g. business suits), as well as speaking in contemporary British Accents.
In the 2012 all-female ChalmersspexThe Brothers Lumiére, this was Played for Laughs. The brothers are trying to create a film, and have employed a scriptwriter to write it. All his ideas seem...somehow familiar.
In one production, the anachronism was boosted by having actors dressed as Hagrid, James Bond and Gandalf come on in the background as the ideas were mentioned.
Some modern productions of Hedda Gabler(1890) do this, particularly with the clothing styles.
In Flashdance: The Musical(set in 1983), at least the North American touring production, the Coca-Cola machine in the lunchroom is a 1990's model.
Assassins has characters interacting who were not alive at the same time. For example, John Wilkes Booth died in 1865, while John Hinckley was born in 1955, but they're in some scenes together. Booth also has a good deal of interaction with Lee Harvey Oswald, born in 1939. Meanwhile, Charles Guiteau (died 1882) tries to romance Sara Jane Moore (born 1930).
The number "Cool, Cool, Considerate Men" refers to right/left-wing politics with the refrain "to the right, ever to the right." Right/Left would not be a concept until the French Revolution. They are also referred to as conservatives when all the men in the Congress were liberals in the classic sense; contemporary conservatives wouldn't have even acknowledged an unauthorized congress. (The song also starts with the opening bars of the Star-Spangled Banner, but that one is clearly artistic license.)
Adams and Franklin waltz with Martha Jefferson in "He Plays the Violin." In the 1770s, the waltz was considered quite scandalous, what with the dancers having their arms around each other and all.
Anatole Kuragin has a plume of bleach-blond Anime Hair. His sister Helene has short loose curls pulled into a side ponytail.
There's a rave at a 19th century Russian dining club.
The characters attend an atonal postmodern opera.
Dan Savage's Miracle!, set in The '90s, features Khia's "My Neck, My Back", which wasn't recorded until 2002.
Matilda originally had the line "Harry Potter? What a rotter." when Mr. Wormwood is throwing away books during the "Telly" song. In the US production, it was changed to "Charlotte Brontė? Do not wanty.", thanks to a policy of avoiding references to events after 1988, the year the book was published.
This trope is used to great thematic effect in Hamilton: Lin-Manuel Miranda initially conceived of the idea when he realised that Hamilton's story perfectly fit the structure of a typical hip-hop rise to fame and then fall storyline, which made all the more sense given that Hamilton was practically defined by his use of the written word. And so came the idea - to portray this most of American stories, as it was in the past, but in the style of today, using the music of today and played by actors representing the diversity of America today. As a result the musical weaves back and forth between the historical and the modern in everything from the lyrics to the costuming (for which the main actors were dressed up in historical style from the neck down but however's natural for them from the neck up, and the back-up dancers were given altered versions of historical clothing).
All over the place in Reduced Shakespeare Company productions, but especially egregious in William Shakespeare's Long Lost First Play (abridged), where much of the modern-day references are actually supposed to have been written by Shakespeare (other plays acknowledge they're being added by the troupe).
In the stage musical of The Little Mermaid, during Scuttle's introduction, he announces "Airspeed, check! Altitude, check! Landing gear, check! Clear the runway!", despite the story being set about a century before the advent of airplanes.
Lizzie is set in 1892, but the music is largely punk rock. In most productions, second act costumes also draw heavily from these elements (but act one's visuals almost always stay firmly in the 19th century).