A Song of Ice and Fire vs. Sword of Truth: Unsurprisingly given both are very dark takes on High Fantasy but the former is known for it's Grey and Grey Morality and favouring gritty realism over magical solutions, while the the latter has extreme Black and White Morality and relies heavily on magical Deus ex Machina. The animosity is mostly limited to the ASOIAF fans, who have made a long-running game of mocking the SoT series, its author Terry Goodkind, its fans, and its similarity to Ayn Rand's Objectivist philosophy (it's notable that George RR Martin is socially and economically quite left-wing). Animosity on the SoT side is mostly limited to reactions against this game.
Some fans of The Dresden Files never pass an opportunity to point out that Harry Dresden would utterly destroyHarry Potter in a fight. Never mind the fact that the only similarity between the two characters that even warrants any comparison at all is that they're both wizards named Harry.
There was some animosity between Harry Potter and Discworld fans due to a misconception that Discworld character Ponder Stibbons copied Harry Potter, or the other way around. In reality, Terry Pratchett has said that J. K. Rowling is a friend of his. (And besides, Ponder appeared years before.)
Not to mention that the only thing Ponder and Harry have in common is that they are both students of wizardry. And, like 68% of the UK population, wear glasses.
In addition, Sir Terry made some comments criticising an interview with JK that some Potter fans interpreted as being critical of her. Similarly, his jokey line about "defeating evil by throwing a piece of expensive jewellery into a volcano" was seen as a "swipe" at J. R. R. Tolkien, whom he has praised to the rafters elsewhere. Pratchett's fans expect him to be ironic, even about things he likes, but those who aren't used to his style can take offence. And Discworld fans, who strive to be equally ironic, don't always help.
When both Harry Potter and Lord of the Rings had their first movies within a month of each other, many comparisons were made, with small skirmishes between the two resulting. The two mentor wizards (Richard Harris and Ian McKellen) even sniped at each other in the press.
When the first Potter film came out, FoxTrot did a week's worth of strips with Jason, already established as a Rings fan, going up against Eileen Jacobsen, who turned out to be a Potter fan. (The last strip revealed that Jason secretly dressed up as Harry and went to the movie anyway.)
Thanks to Orson Scott Card's fairly harsh criticism of Rowling for her lawsuit against the Harry Potter Lexicon, a few fans of Ender's Game have turned against it too.
There was a bit of a rivalry with The Inheritance Cycle, but that died out when people noticed that it was entirely manufactured by the marketers.
The polarizing series Twilight has inspired quite a few fandom rivalries.
There's Potter vs. Twilight. Never has such a fandom rivalry been so whipped up by the media itself. Twilight, like Potter, is something of a literary phenomenon, and its movie stars a Harry Potteralumnus, so the two were constantly compared. They really have nothing in common though, apart from both being children's/young adult's fantasy series.
A more interesting fact is that the Twilight vs 'Harry Potter war was whipped by the media constantly comparing the two, but then spread to the fandoms themselves on it own accord and media fanning the flames to that. Harry Potter fans stated that a war between good and evil was a story much deeper than a teenage romance, the Twilighters claiming their series is "sexier and more mature than HP (sic)".
Made worse, when asked "who would win" with Harry versus Edward, Stephenie Meyer didn't just say "Edward would win due to having Super Speed", she said "Edward would kill Harry because his Super Speed makes him faster than wizards shooting wands at each other. You can't say something like that without Internet Backdraft, folks.
To a lesser extent, Buffyverse/traditional vampires/Anne Rice fans. They seem to imply that a true vampire genre fan could never like the blasphemy of Twilight or the take on the female lead when Whedon created such a strong female. However Whedonesques are usually very civil on expressing their dislike of the saga, while not bashing it or its fans.
Harry Potter fans joined themselves to avoid the Eclipse movie from trending on Twitter on the day of its midnight release.
The Hunger Games fans are almost always sided with Potter. There are even "District 9¾" alliance pages on Facebook. Some are former Twilight fans, as Stephenie Meyer's rave review brought The Hunger Games to the attention of many, frequently resulting in readers realizing how much better The Hunger Games is than Twilight.
And then, of course, we always have that small group of pretentious elitist snobs who believe that kids should instead be reading Carroll and Twain and that adults should be reading Austen and Faulkner.
On the Narnia fans' side, it was primarily caused by the author of His Dark Materials loudly criticizing the Narnia series. You can't say stuff like that without causing a fandom war, folks.
For some time there was an entirely mock rivalry between Isaac Asimov and Harlan Ellison®, who in real life were close friends: Ellison brought it to an end, because his regard for Asimov was so great he was worried fans would take the rivalry as real.
Asimov had mock rivalries with several writers. He once wrote that he and Lester del Rey had been playing a friendly (no, really) game for decades, wherein one of them would "insult the other. The proper response to an insult is a slur, which must then be topped with a minimum of delay." Their wives would keep it from escalating to Global Thermonuclear War by making them stop by the time it got to three exchanges.
Asimov and Arthur C. Clarke reached an understanding in which Clarke would admit he was only the world's second best science writer and Asimov would in turn admit he was only the world's second best science fiction writer.
Fans of Jane Austen vs. fans of the Bronte sisters. Charlotte Bronte personally never liked Jane Austen's work herself, and many fans of Austen's romantic comedies and the Bronte sisters' Darker and Edgier Gothic romances dislike the other's works.
Lit Fic fans vs fans of any genre based writing for that matter.
Fans of the classic books vs. fans of popular contemporary books.
Battle Royale fans (book, movie and manga alike, who also have their own rivalries) vs. The Hunger Games fans. Fans of Battle Royale frequently accuse Suzanne Collins of having ripped it off in writing The Hunger Games, an accusation that HG fans counter by noting that BR's plot itself isn't all that original (see ]] The Running Man, The Long Walk and Gladiator Games in general). Moreover, Collins herself has stated that she had never heard of Battle Royale until her publishers brought it up to her, and the inspirations for The Hunger Games were the legend of the Minotaur in the Labyrinth and channel-surfing between horrifying news of war and superficial reality television.
Within the Cthulhu Mythos, some fans prefer H.P. Lovecraft's original inscrutable, human-morality-means-diddly-squat vision of Cosmic Horror Story, while others have no problem with August Derleth's belated addition of good-vs-evil and four-element themes to the Mythos. Even the "Lovecraft Circle" itself had members who'd argued over this one.
There's a small degree of this between Animorphs and Everworld, due to the fact that K.A. Applegate (the husband-and-wife team of Katherine Applegate and Michael Grant) left to work on Everworld, the Animorphs series was largely ghostwritten, and suffered because of it. Fortunately, nothing gets too overly heated.
Fans of 1984 vs Fahrenheit 451. Fans of 1984 criticize Fahrenheit 451 as being a knockoff due to similar themes such as censorship and a dystopian government advocating anti-intellectualism and criticize the book's sci-fi elements. Meanwhile, fans of Fahrenheit 451 criticize 1984's length and writing style.