It just had to be the route of Fate/stay night which has Saber calling Archer and Rin intimate, Archer putting his jacket around Rin to keep her warm, and a lot of other Subtext that also brings us Archer's Face–Heel Turn and Rin hooking up with Shirou.
Iris from the third game gets teased with Phoenix. She once was in a relationship with him while impersonating her sister, Dahlia, and Phoenix once put his life on the line to protect her by drinking what might be poison. Ending involves a scene when Iris is happy about Phoenix looking at her and Phoenix is slapped for it by Pearl(who is die-hard Phoenix/Maya Shipper).
The fourth game seem to be teasing Apollo/Klavier with Klavier's very first line! ("I'm used to being inspected by the ladies... but I admit this is the first time I've felt this way with a man!")
Juniper VERY clearly harbors a crush on Apollo, to the point where she will start knitting a scarf full of hearts while gushing about him. Some have joked about how Juniper is Vera 2.0, partially because of this. (Apollo himself seems oblivious to all of it, however.)
Apollo and Athena also have a fair deal of it - small things like Klavier assuming that they are a couple fade in comparison to how the second and third cases are all flashbacks as Athena wonders why Apollo left the agency, and the fifth case... Apollo wasn't wearing that eyepatch due to the courtroom bombing, but to stunt his perception powers on purpose so he wouldn't distrust Athena due to her tells... and then, when he is given more concrete reasons to distrust her, he gets on the witness stand and begs Phoenix to prove him wrong. "How can you have faith without doubt?"
Athena also has quite a bit of this with the merciless prosecutor Simon Blackquill, specially in the fifth case when her and Blackquill's shared Dark and Troubled Past is revealed and the players learn exactly how important they actually are to each other... Blackquill is determined to protect Athena, the daughter of his dead mentor, even if that means sacrificing his own life to do so.
From the second Edgeworth game, we have Raymond Shields and Katherine Hall, a defense attorney and his client. They show the same kind of trust in each other you see from Phoenix and Maya, and Gregory mentions from their first meeting that he told Raymond to 'note', not 'dote'. Raymond won't hesitate to suspect Katherine when all evidence points to her, however.
The licensed game adapation of Tim Burton's film Alice in Wonderland (2010) takes ship teasing in a completely different direction (see the Film section above). While the movie contains hints of Hatter/Alice, the game contains hints of Hatter/White Queen.
Riki, the protagonist of Little Busters!, obviously has ship tease with each of the girls he can end up with. But more notably, he also has a veritable shitton of it with his Childhood Friend Kyousuke. Much of it - the touchiness, their tendency to get into suggestive situations together - is simple Ho Yay, but in one optional scene Riki outright admits mentally that he has feelings for Kyousuke but isn't doing anything because he doesn't want to ruin their friendship, and everything we hear from Kyousuke on the subject is ambiguous at worst and very much implying he feels the same at best. However, since there is no Gay Option, not even a comedic one like in CLANNAD, the pairing is sadly never to be. At least, not in any of the realities the player can see, anyway.
Refrain takes it even further: their last meeting before the dream world crumbles and Riki returns to the real world has Riki completely distraught, yelling that he likes Kyousuke and wants to be with him forever, and Kyousuke screaming back that he likes Riki so, so much more, all coming across in exactly the same way an Anguished Declaration of Love would. And then in the end when Rin asks Riki who he likes, we never hear a response. There's nothing that's completely incompatible with an interpretation that they're just very strong friends, but if they weren't, this is sure as hell what it would look like.
In Dangan Ronpa, there is a little bit of it between Naegi and Maizono, due to them being classmates in middle school and the only two who knew each other before Hope's Peak Academy. It all comes to nothing, as Maizono ends up dying after trying to kill someone else and frame Naegi for it. However, she leaves behind the name of her killer, written in blood. Kirigiri speculates that this may have been in repentance for her earlier manipulation.
After Maizono's death, hints of Kirigiri/Naegi started popping up. What with both Monokuma and Asahina mistaking the two to be dating when they were meeting up secretly in the sauna, Kirigiri being furious after Naegi keeps a secret from her (when early on he convinces her that they should trust each other completely with their secrets), and Kirigiri coming to Naegi's rescue in the garbage dump after his execution failed, it's not hard to assume that there's something going on between the two.
In Super Dangan Ronpa 2...oh, man. There's no question that Kuzuryuu and Pekoyama love each other very much. The real question is if the love is romantic or familial in nature. Pekoyama, Kuzuryuu's bodyguard, is loyal to him to the point where she kills one student and tries to condemn the others (including herself) to death just to get him off the island, and encourages him to use her to any means he desires. Kuzuryuu, normally a chilly jerkass, breaks down crying at Pekoyama's sentencing and risks his life in a futile attempt to save her from her execution. Plus, he actually refuses to treat her like a tool and sees her as a beloved companion, which is in marked contrast to his behavior towards his other classmates. It may be the emotionally charged atmosphere of Chapter 2's trial, but there are a lot of fans who believe there's something romantic going on there.
Pekoyama's Dangan Island route outright confirms that the reason why Hinata couldn't end up with her is because she already likes someone else, with that someone else heavily implied to be Kuzuryuu. A Valentines Day tweet reaffirms this notion. Also, in chapter 6 the reason why Kuzuryuu eventually decides to choose the Forced Shutdown option is because Pekoyama would have wanted him to, otherwise he would have chosen the Stay option just so he could be with Pekoyama again.
Gundam Tanaka and Sonia Nevermind probably get the most ship tease second to Pekoyama and Kuzuryuu. Not only is Sonia the one character to make Gundam blush, she is also the only one he doesn't refer to in a demeaning manner and clearly respects her the most out of the class. It goes as far as him referring to her as his "Dark Lady" and in one of his free time events, he tells Hinata about his fantastical image of her. On Sonia's end, she greatly admires Gundam and his hamsters, and even joins in on his antics from time to time. It's to the point where it seems that she's the only person who not only accepts Gundam's fantasies but also understands and believes them. The most apparent ship tease occurs in Chapter 4, where Sonia tagged along with Gundam whenever the group split up, the two of them showed great team work and deduction during the trial and Sonia defended him to the bitter end when he was accused of being the murderer.
There's also quite a bit ship tease between Hajime and Chiaki. During one of her free time events, she compliments him on being well-built and says that she trusts him enough to carry her, all of which leaves Hajime very nervous and self-conscious. He even mentions his heart racing on more than a few occasions with her. During her final free time event, Chiaki admits that she doesn't know a lot of things, like how to fall in love with someone, and asks Hajime if he'll keep teaching her. She also asks him not to never forget about her after he gets off the island. He promises that once they escape, he'll take her to all kinds of different places.
It helps that all the couples mentioned here (and in the first game above) are constantly paired together in official merchandise.
Word of God states that after the first game, Junpei chased after Akane for his whole life. The sequel shows that Akane genuinely wanted to be with Junpei, it was just that she had to devote her life to saving the world from Radical-6. One of the writers confirmed that she was not lying, which lends credence to the theory that her flirtation with him wasn't as pragmatically motivated as fans thought. In VLR, Junpei's future self Tenmyouji is embittered by the whole experience, but both Akane and an alternate version of Junpei are confirmed to appear in the third game, so who knows where they'll end up?
While not nearly as obvious as with June, Junpei also has some Ship Tease with Clover. On the route to the true ending, they share a couple of intimate moments, which aren't romantic in and of themselves, but do pave the way for what can be read as Clover developing a bit of a crush on Junpei.