Rosa is usually ignored or looked over when it comes to Kain/Cecil. She is also often poorly treated by Cecil/Rydia shippers (those who don't Take The Third Option in form of OT3, that is).
Edge is heavily antagonized by both Kain/Rydia and Cecil/Rydia shippers. Sometimes Played for Laughs, but sometimes it's serious.
Izayoi from the sequel, due to her being a woman in Edge's group of personal bodyguards, and thus making her quite close to Edge. Plus, her dialogues with Edge is quite Ship Tease-y. Edge/Rydia fans are not pleased, and Izayoi is either ignored, antagonized, or paired off with other guys (usually Gekkou or Kain).
After more than seventeen years, bothTifa and Aeris are still prone to being demonized and/or killed by shippers who favor a different pairing. Even more disturbing is the fact that Aerith is canonically dead by the end of the game anyway, and by Word of God that is irreversible.
Lucrecia, who is Vincent's canon-designated love interest, is often the victim of this by fans of other pairings involving Vincent (most notably Yuffentine and Valenwind) since it is quite obvious that, even though they have been separated for decades, he will never be involved in another relationship unless they get her out of the picture somehow. This is rather literal when fans actually try to write her off as being dead despite Word of God statements that she IS still alive (and thanks to Jenova Cell, CANNOT die), and also less literal in that fans will villanize her just for the sake of justifying Vincent "moving on." What makes this whole situation even worse, though, is that Dirge of Cerberus - as one of its many Retcons - portrays Lucrecia as being far less morally ambiguous than she was in the original game, which leads to the suspicions that the designers are just deliberately messing with the fandom.
The Final Fantasy VIII fandom is rife with shipping fans united only by their hatred of Rinoa for getting Squall, and a raging desire to see him hooked up with anyone else. The majority of this is focused on Quistis or Seifer, but other shipping ranges from the odd (Selphie, Fujin) to the even more Ho Yay (Irvine, Zell) to the incestuous (Laguna/Squall), to the bizarre (....Cerberus/Squall? MY EYES).
Said Rinoa haters seem to be especially fond of the fandom's favorite Epileptic Tree, which is that Sorceress Ultimecia is actually Future Rinoa. It tends to come up with disturbing frequency in shipping wars, even though Word of God has disproved it.
On the other side you have the fans who strongly hate Squall and love Rinoa (although there are fewer of them than the Squall-lovers) and who want to see her with anyone but Squall. They tend to paint Squall as uncaring or cold toward her, sometimes even an outright cheater, which sets Rinoa free to be with a "better" character — most often Quistis or Seifer, sometimes Irvine, Selphie, or even Ellone.
There is a small but intense group of Final Fantasy XII fans who are quite polarized between Balthier/Fran and Balthier/Ashe as the 'true' canon pairing (despite the fact that there are no real canon pairings in the game) — this is complicated by those who think Balthier is probably batting for the other side, which in itself is odd considering that he never makes any advances on the guys. Vaan is also on the receiving end of the fan-hatred when it come to pairings for the original game but that seems to have more to do with him being characterized ratherpoorly in it in contrast to the others, as no such dislike exists among the Revenant Wings shippers (in which Vaan was actually allowed to take the center stage and develop).
Alyssa from Final Fantasy XIII-2 is hated by Hope/Lightning fans for supposedly getting in their way. Note that Hope never shows any romantic interest in Alyssa, Lightning, or anyone for that matter. Funnily enough, one alternate ending has her getting thrown in prison for her actions, and in another you actually can remove her from the timeline.