Harry Potterfanfic set in the Marauders era tends to ignore the fact that it's set in the 1970s. James, Sirius, and the others will merrily chat on their mobile phones and use computers (both of which did exist in the 1970s, but not in the form we're familiar with today and weren't common in any case, to say nothing of the anti-technology field of Hogwarts), send text messages (which actually weren't invented until the late 1980s or popularised until the late 1990s), listen to 1990s or 2000s music, and watch recent films.
Not that Fan Fiction set in the "modern" time period is any better. While granted it is easy to miss, a lot of writers forget that the series is set between 1991 and 1998, and a lot of the things they have the characters reference or possess shouldn't even exist yet.
And let's not even get started on My Immortal, much of which supposedly takes place in the 1990s. There is no noticeable difference between the "past" and "present" portions of the story, especially not where music history is concerned.
And then there's the Hogwarts Founders fanfic, where circa-1000 A.D. Scotland is often rendered as a mishmash of High Medieval, Renaissance, Victorian, and pure fantasy elements.
There was one in which Ravenclaw went into exile in America, which (unless she was a Viking, possibly) would have been impossible even if you discount that she apparently got there on the train.
There's another one where Ravenclaw and Hufflepuff are at a pre-Hogwarts British Boarding School — in different Houses — and Ravenclaw calls Hufflepuff up on the telephone to make plans. At least they didn't have cellular phones?
And one where the founder era has showers, paper, tea and coffee, meetings at 15:25, springy metal net beds invented in the 1940s. The usual stuff. What's more, Hufflepuff is supposed to have been born and raised in a Finnish manor house. Tasty, as the area was then still in Iron Age, and entered history in the 1300s.
This fanficrants post describes a Twilight fic in which the writer tries and fails badly to avert this, by having Carlisle listen to the radio rather than play video games. In the 17th century.
The Umineko: When They Cry fan fic "Witches & Woodlands," which takes place during the year 1986, brings up so many anachronisms that the characters are unable to ignore it: for example, one character mentions reading forums on the Internet despite that not being common practice for another decade or so; another mentions how people will behave when playing MMORPGs once they're invented, and what games will be like in the future. They even mention anachronisms within the actual series itself through their Medium Awareness, particularly when one character dressed up as a character and played a song from Touhou, which was first released in 1996; said character claims to have made the costume and song herself and plans to sue the companies behind Touhou for plagiarism. In the end, everyone agrees that all the anachronisms can be explained with some kind of retarded logic.
The Kantai Collection fan fic Ambience: A Fleet Symphony combines this with In Spite of a Nail. The fic is set in 2029........a post-apocalyptic 2029 that is, as a result of World War III, which occurred back in 2010 and saw nukes being chucked around, with the US and certain areas of Europe and East Asia being hit the hardest. In spite of all this, there's references to goods, weapons, and other things that were introduced after 2010, and because of the timeline divergence, these all seem out completely out of place.
Sometimes Axis Powers Hetalia fanfictions may have technology existing at a time it should not. However, it seems more common for authors to put groups and organisations of nations in the wrong time period; two examples that have happened at least once are the WW2 Allies having tactics meetings in the twenty-first century (although the rest of the story made it obvious that the writer at least knew the war was not still happening, so it's unclear what they thought the meetings were for), and G20 meetings occurring at the height of the British Empire.
In the Yu-Gi-Oh! fanfiction Decks Fall Everyone Dies, the characters write a propaganda play (to revive card games as a form of government) set in ancient Egypt. The play, according to the scenes depicting the rehearsals, involves Ho Yay, thrift shops at which the penniless duelist buys the Pharaoh's old clothes, Riverdance, Rap music, and holographic disco balls.
The Infinite Loops has Loopers pinching random items from other universes whenever they can. This can lead to some rather mishmashed tech levels.
Meta-subversion: All 1632 fanfic writers who want to have a chance of getting past the critics on the Baen Books Forum have to remember that "up-time" is 2000AD, not "today". Anyone making references to flatscreen monitors, the War on Terror or Windows XP will get slapped down hard. The series proper averts it though. In an Alternate HistoryTime Travel story, deliberate anachronism is sort of the point.
Warriors of the World is set in an unknown time period, but the setting is clearly medieval with its houses of stone and wood heated by fireplaces. Yet clothes have zippers, there are proper showers, someone's wearing non-functional headphones as an accessory and everyone knows what it is, and corporations called as such exist.
The Contractually Obligated Chaos series sort of runs on this. There are frequent mentions of things, particularly the Running Gag of characters using Tumblr to keep an eye on the plot, which didn't exist in the time period when the show on which it's based was in syndication, nor in the years immediately following, which is when these stories would theoretically be set. The 'timeless' nature of the cartoon, however, allows for a massive Time Skip to bring it into the 21st century; either that or it's a case of MST3K Mantra. Take your pick.
In Real Life, April Fool's Day has Catholic origins dating as far as the 14th century. In An 'April Fool's' Errand, it started in Ancient Rome in dedication to Hades (or at least to the fictionalized accounts of him, Persephone and Demeter told by Homer).