Samurai Champloo opens the series with a title card declaring that it is not a historical document. It then gleefully throws everything it can get its hands on (from hip-hop to baseball) into the Edo period of Japan (1603-1867/1868). Doubly amusing because baseball is a hugely popular sport in modern Japan.
" (clears throat) This is not an accurate historical portrayal. Like we care. Now shut up and enjoy the show."
One of the first villains seen in the show is a guy with dyed blond hair, several facial piercings, wearing a tracksuit. It goes down/uphill from there.
Slayers has most of this in the form of the outfits some of the cast winds up wearing, mostly in the gag episodes. Episode 16 of season 2 involves the cast getting involved in a game similar to tennis; several wind up wearing sportswear that sticks out from the Medieval European Fantasy outfits like a sore thumb.
There's also a vehicle that operates like a more modern train in the third season, but it looks like a giant...thing made of stone, so it might not count.
Soul Society in Bleach appears to resemble Edo-period Japan in clothing and architecture. But the Shinigami use cell-phone like devices, have a highly advanced research division (complete with Mad Scientist), and several characters sport sunglasses or other modern attire.
Which is understandable — Edo period stuff but they can visit the modern world. Just when you become okay with it you get a flashback with computers and sunglasses around the turn of the 20th century Earth time, with Jazz, in Edo period Soul Society. Note that the first Jazz recordings date to 1917, and that sunglasses started being mass-marketed in The Thirties.
An omake in the volume the chapter appears in has the author telling the character (Shinji) that, naturally, jazz didn't really exist at that time period in the real world, cue the comical bafflement of the jazz-loving character who must be wondering what on earth he's listening to if it doesn't exist. At that time period, modern jazz may not have existed, but its very early genesis was already beginning to flourish so... it kinda did exist, after all.
D.Gray-Man has a slightly less frequent occurrence, which is anything related to Komui. While most of the technology seems fairly well-depicted for a series taking place in the late 19th century, it is unclear where exactly Komui got his hands on hover-devices and the technology to build Komurin. His standard toolset consisting of a giant electrical drill and some other power-tools are also pretty advanced for the setting.
General Cross dual-wields modern handguns that shoot magical homing bullets.
And where exactly did Komui get his hands on modern clothes while everyone else wears 19th century clothing?
This is actually commonplace for the tech division. One can only wonder why a 19th century religious organization working under the Vatican would have a tech division in the first place...
The places that the cast of Soul Eater go to are... varied. Medieval Japanese villages with Assassin problems, Polish villages who specialize in Golem manufacture, mixed with modern depictions of Venice, Italy and an apparently modern American neighborhood., and London (well, Tower Bridge, at least). Also, The Grim Reaper and his students all live in a city in Nevada.
In Gintama, aliens (known as Amanto) forcibly opened up Japan instead of Commodore Perry and crew, bringing all sorts of new-fangled technology to Edo (space travel, electric fans, bazookas, etc). And since Gintama is supposed to be a Gag Series, you get things like the main character being a big fan of Weekly Shonen Jump (most notably Bleach, since he uses a sword too), idol singers, and countless references to modern pop culture mixed in with more traditional fare, like The Shinsengumi, the Jooi resistance, and the Oniwabanshu (though disbanded in the series).
Black Butler is set in London in late 1888, when Jack the Ripper was at large. The maid in the house washes clothes with a washing machine and laundry detergent sold in a box, the chef cooks food with a flamethrower, badly, and Jack the Ripper fights with a chainsaw.
And there are cell phones used by people who look like they're in the mafia.
As the manga's carried on, the anachronisms have been toned down, mostly being either aesthetic (women in not enough clothes, and so on) or difficult to fix due to plot (allowing there to be only five or six servants on an estate makes maintaining The Masquerade easier, etc.). The remaining offenders are the Reapers, who, being otherworldly, seem to have access to modern technology such as plastic-framed glasses, a modern wristwatch, and the aforementioned chainsaw.
Osamu Tezuka loved to throw in gross anachronisms into his historical works. The first volume of Phoenix, for instance, has an ancient Japanese general leave to read a James Bond novel (which may be a Woolseyism on the part of the translator), and things like televisions and refrigerators are worked into other volumes of the series via Bamboo Technology.
Lampshaded in Dororo. The titular thief compares himself to Nezumi Kozō (a folk hero along the line of Robin Hood), then points out that Nezumi Kozō hasn't even been born yet. This is in addition to numerous straight usages.
Near the end of the Buddha series, Buddha heals Prince Crystal by placing his finger on the tumor that is killing him. One of the prince's advisors said he heard of this power before, and asks Buddha if he's E.T. A few chapters later E.T., Yoda, and Cherry (from Urusei Yatsura) make a cameo appearance.
Occurs in both Fate/stay night and Fate/Zero. The servants tend to be anachronistic, with most being summoned wearing very modern-looking clothing and hairstyles, and Saber and Gilgamesh wearing armour (and in the latter case, a lot of various weapon types) that did not exist at the time their legends occurred.
Oh! Edo Rocket has a lot of this. Supposedly set in the early 19th century, but shows various characters using modern technology such as computers, TVs, and pocket calculators. They also often use terms that weren't used in their time period. Example: one city commissioner calls another a "bleeding-heart left-wing liberal", which the accused liberal then proceeds to lampshade and Break The Fourth Wall simultaneously by saying, "Now, sir, that term wasn't used in this time period." The other man replies, "I'm sure the audience understands what I mean."
One Piece, presumably set sometime in the 15th century, features technologies such as radio, video, submarines, steam engines, surgery, and a wide variety of electronic machinery. Radio and video are lampshaded with the Rule of Funny - the signals are transmitted via handheld snails called Den Den Mushi. Yes, snails.
Nausicaä of the Valley of the Wind, as mentioned in its page, has sword-swinging cavalry charges usually being backed up with machine gun fire from levitating bell jars. Torumekian gunship pilot uniform consists of full medieval-style plate armour complete with a spiked visor helmet. The setting of the movie is post-apocalyptic, so it is less odd than it sounds.
Certain Gundam series tend to blend archaic aesthetics with space age society and technology. For example, most One Year War themed series feature World War II aesthetics, ranging from Zeon uniforms being Wehrmacht knockoffs to Federation ground troops using full sized walkie-talkies and backpack sized communication gear in the field, while later UC series (as well as Gundam Wing) tend to place emphasis on European fashion and cultural themes (i.e. the guillotine in Victory Gundam) in spite of their advanced technological settings.
In-universe example: An early episode of Pokémon featured a carving of a Mewtwo on one of the external walls of what appears to be a century-old lighthouse. Mewtwo wasn't even created until the second half of the Kanto arc.
Fullmetal Alchemist is essentially set in a Fantasy Counterpart Culture version of the early 20th century. Everyone wears modern clothing, though occasionally you'll see characters wearing time appropriate clothing (especially if they're older). Certain places are less technologically advanced than our early 20th century, but they are also more advanced than our counterpart years; for one they have "automail", which is even more advanced than our current mechanical prosthetics.
Naruto is a weird example. They have legitimate ninjas and have a society that's very reminiscent of older times in Japan, but they have more modern things like modern chain-link fences, sunglasses, the characters tend to wear more modern clothing, and at one point some characters use a VCR. They also seem to have much more modern-ish hospitals.
Lampshaded several times in the Rock Lee spinoff when Kabuto questions whether certain things like electricity exist in this world. Which is a contradiction since in the original series he uses some kind of medical computer at one point.
Word of God says the Naruto universe is modern except for their lack of cars and guns.
The Dragon Ball series is full of this. Technology is so advanced that objects larger than houses can be stored in a tiny, pill shaped capsule. On the other side of the spectrum the world outdoors is filled with saber tooth tigers and dinosaurs.
Tono To Issho is set in the Sengoku Period, but some characters wear modern clothes, and apparently anime and microphones have been invented.