Misfile contains a rare example of a ship fighting itself. Ash/Emily shippers all want to see Ash and Emily hook up, but because Ash is a boy turned into a girl they can't agree on how the comic should end. One side desperately wants Ash to decide to stay a girl, the other wants to see Ash back as a guy. The fighting hasn't yet reached dangerous levels, but the arguments crop up with regularity.
While the MegaTokyo fandom as a general rule tends to be quite civil, there does seem to be a distinct undercurrent of dislike between fans who think Piro should stay with his current girlfriend, Nanasawa Kimiko, and those who would like to see him get back together with his previous girlfriend, Tohya Miho. One can generally tell which girl a particular fan supports by whether he refers to Kimiko by her given name, or by her familly name (Nanasawa).
Sluggy Freelance has a number of One True Pairings. Torg is most often paired with Zoë, Gwynn, Sasha, or Aylee (the alien), and Riff with Gwynn, Sasha, or various insane ladies.
Homestuck has this, although considering the series' romance is Love Dodecahedron it's not all that much, fandom attitude having finally given in to Ship All The Things, Fill All The Buckets, and Threeway Fixes Everything.
The main two locked standing in an eternal bloodbath are Dave/Terezi and Karkat/Terezi. It doesn't help they've both convinced their side is solid, set-in-stone canon when neither one actually is. The rest of the fandom suffers many fights, (John/Karkat vs Dave/John or John/Vriska, PBJ vs anything that stands in its way), but nothing touches the epic bitchfight of Terezi's flushed quadrant.
It's enough that it's become an increasingly popular trend to Take a Third Option and ship Dave/Karkat- which gets a lot of general hate.
As of the 4/22/15 update, Dave/Karkat is implied to have happened in the new timeline, making this a canon case of a Take a Third Option same-gender couple. However, there are still enough shippers of Dave/Jade to keep the wars going. Near the very end of Act 6, Dave/Jade gets teased, and in the new universe Karkat, Dave, and Jade are seen together having a picnic.
Not to mention the sub-Love Dodecahedron of Equius-Aradia-Sollux-Feferi-Eridan, though its fighting is relatively tame. Sollux/Aradia, Sollux/Eridan, Eridan/Feferi, Equius/Aradia, and Sollux/Feferi definitely skirmish some [in different fight variations], but seemed to have stopped actively trying to kill each other.
Rose/Kanaya and John/Rose are often pitted against each other in spite of the latter's non-importance to the story. When the official merchandise released a fan created design based on the latter, the comment section and associated Tumblr tag were swamped with hate.
It's also worth mentioning that the fact that there are 4 ways to ship, 3 requiring two participants and one requiring 3, which doesn't help, only contributing to in-ship fighting [red quadrant vs. black quadrant mainly, but also with red vs. black vs. pale vs. ashen [featuring another character].
There were many reactions to Paz and Kat dating in Gunnerkrigg Court, ranging from "This comic has lesbians in it! Ruined Forever!" to "They're so cute together!" But to the fandom's credit, most complaints were about Kat getting with Paz instead of Annie when Kat's friendship with the latter is a focal point of the comic.
El Goonish Shive: With the raising Ship Tease between Elliot and Susan, there have been mixed reactions. Elliot is the perfect candidate as Susan's potential love interest, since he literally "fails at perversion" and Susan's looking for a good man. But Elliot is already in a relationship with Sarah, Susan's best friend, so she's in risk to fall into a Yoko Oh No situation. In-universe, Sarah decides to break up with Elliot shortly before Elliot realizes she's like a sister to him, and she has no problem just handing him off to Susan (though she doesn't think it's likely).
After that situation is resolved, the new combat seems to be Elliot/Ashley vs. Elliot/Anyone Else, as he meets her, decides to date her, and finds that she's a great fit for him very quickly, leading to arguments over whether the relationship is an adorable breath of fresh air amidst tons of romance drama or forced, over-perfect, and boring.
A literal version happens in MoringMark's Star vs. the Forces of Evil ongoing fancomic, appropriately titled the Ship War AU, since the premise revolves around Marco and Star's future daughter and Marco and Jackie's future son travelling back in time to make sure their respective parents get togethernote And in Jam's case, prevent an apocalypse, and thus they end up fighting each other.