Arc Fatigue: While nearly all fans agree that Miyu's brother is indeed awesome, some feel Drei's story arc exploring his and Miyu's backstory has overstayed its welcome. Having gone on for 10+ chapters with any final resolution at least several more chapters away, it's not hard to see why (by comparison, the entire original series ran for just 13 chapters). Making this even worse is the fact that the general details of the story are a Foregone Conclusion.
Thankfully, the flashback is over, and Shirou even lampshaded how long it was.
Alternate Character Interpretation: The Ainsworths, especially Julian. Are they a group of Well-Intentioned Extremists or sociopathic monsters willing to abuse a little girl for a plan that might not even work? While their backstory and explanations point heavily towards the former, some of the actions they do are without a doubt the second. Some fans may sympathize with them, but others share Alternate!Shirou's desire to kill them as slowly and painfully as possible.
Broken Base: Illya's extremely bratty behavior towards Kuro in Zwei turned quite a few fans against her, even those who didn't like Kuro to begin with. Most of the fans her actions upset will point out that Kuro wasn't really hurting anyone, and pretty much everyone except Illya seems to be okay with her presence, making Illya's frequent physical attacks Disproportionate Retribution (especially since Kuro's curse means she can't fight back). Fortunately, she does get better after narrowly stopping Miyu and Kuro from fighting to the death, and the intervention of her mother also helps a lot.
Ensemble Darkhorse: Even if some fans believe his arc has overstayed its welcome, it almost unanimously agreed that Alternate Shirou Emiya is made of pure awesomeness. Some have gone as far as to claim him to be one of the best incarnations of the character, rivaling if not surpassing even the likes of UBW and Heaven's Feel Shirou, but even Archer himself.
Foe Yay: Alternate Shirou and Angelica has become popular during their fight. The former has some ambiguously flirty dialog along with directly calling her cute. The latter seems to be continually trying to talk him down from fighting and seemed hesitant to kill him.
Do you remember the time when Miyu said "brother" to Shirou when she was at Illya's home? In the universe from which Miyu came Shirou is her brother and Miyu is the vessel of the grail, and he went on a crusade against (very powerful) people who kept her captive, sending Miyu to the universe Illya was from, but ended up getting caught by them and tortured in a dungeon. And she probably remembered about him every time she saw Illya's Shirou. Ouch.
There is also a scene at the end of the 6th episode of the anime (not present in the manga) where Rin wants to take unconscious Illya home after the Saber card was captured, only for Miyu to interrupt and say that she will be the one to do it. And she carries Illya home while a music box plays in the background. Truly a heart-warming moment, but if you know that Miyu is her universe's Holy Grail vessel and she suffered quite a lot because of it then you know why she did it and while it still is a heart-warming moment then, the context changes from "for a friend" to "for someone who bears the same burden as me".
Really, just about everything with Miyu becomes this when we find about about her world and her life with the Ainsworths. It starts as early as when she asks Illya why she's fighting. She's not happy about Illya treating collecting the cards like a game, probably because of how the cards are used in her world, and by who.
Just Here for Godzilla: Quite a number of non-yuri fans intentionally jumped in after knowing that Gilgamesh appears along with Gates of Babylon, Vimana and Ea.
She's desperately trying to ignore it; like when she accidentally kissed Miyu during a dream of Shirou just as she woke up. Miyu didn't mind.
Illya's Maid Fetish results in her mounting Miyu (Twice!) before she can compose herself. Miyu didn't mind.
Chloe, on the other hand, seems to actively encourage this. To the point that apparently, she spent the entire time between Chapters 4 and 5 of the second season just going around town kissing girls (Including Taiga!). Granted, she was doing it to drain their mana, but judging by the one we actually get to see, she also took the time and effort to seduce at least a few of them first.
Then comes Chapter 8 of Drei.
Chloe in general with both Miyu and Illya. She managed to make the former unconscious with a French kiss.
From 3rei, Tanaka is widely hated for being as stupid as one could possibly get without forgetting how to breathe. It hurts that for her desire to get rid of the Ainsworths, which is the only defining characteristic of her personality, she does almost nothing during the Final Battle, only giving Illya a long distance powerup.
Gilgamesh: Its hard not to see his teasing of Chloe and Illya as flirting. Double points when he does it toward Chloe when he is naked.
This is morbidly hilarious considering his Mainline relationships with their respective characters.
Shirou: It be easier to list people who aren't teased with Shirou though the most subtle probable goes to Sella up until Her own Sister and Shirou's mother are basically trying to pimp her out to him. In the manga, Rin has the most ship tease. Rin has been the protagonist of many side chapters focused on her relationship with Shirou, that no one else has had.