Our hero certainly spends a lot of time rescuing that princess, so he must really care about her. And it's a safe bet that she really cares about him, so it's only natural to assume they're more than Just Friends, right?
But wait...had anyone actually said that the two of them are together, or are you just assuming? Is the character denying the truth or stating it when he says "She Is Not My Girlfriend"? Is it Not What It Looks Like - Or is it? We could ask questions all day.
Either way, in a serial work, this allows the author to gently cover their bums when it comes to either Ship Sinking ("Hey, the ship never set sail in the first place") or Cleaning Up Romantic Loose Ends ("The potential was there from the start"). Expect there to be some fans who pair the couple up regardless - and the writer may state that they Hooked Up Afterwards, or leave it to the reader's imagination. Naturally, this goes hand-in-hand*
Tower of God: Baam chooses to follow Rachel where ever she may go, even if that includes fighting sea monsters even though he had to be taught why people would use violence at all few months prior. Many note that this has a romantic notion to it and the author (of course) hasn't said a thing about their relationship status. She could be a mother figure, big sister, best friend or love interest.
Lucy has multiple potential love interests, the most prominent being Natsu and Loke.
Lyon and Sherry has been busted for Sherry and Ren
Erza/Jellal gets this too. When they almost passed the point of being implied they suddenly backpeddled, so it's still a big if.
Neon Genesis Evangelion has an interesting variation: despite the tons of Unresolved Sexual Tension between Shinji and Asuka, it is heavily implied in Rebuild 2.0 that he chose his half-sister Rei over Asuka (both had a close encounter with an Angel: Shinji didn't even try to save Asuka yet when the same happened to Rei... it turned out to be Shinji's Berserk Button). It is clear that Rei reciprocates his feelings (her monologues; the fact that when Zeruel attacked, she carried his SDAT player into battle instead of Gendo's glasses), yet S/A shippers argue that the S/R ship is not viable due to obvious reasons. A counterargument: any kind of romantic relationship between Shinji and Rei is NOT incest because while she is a clone of Shinji's mother, Rei is also a human-Angel hybrid and as such, dodges any moral barriers regarding the matter. Of course, Your Mileage May Vary; especially since Shinji doesn't have the slightest idea about Rei's origins.
In Pokémon, Misty was this, to an extent, for Ash in the Original Series. It varies between translations - the original writers stated that the two are too young to recognise their feelings, and most of the dialogue implying attraction in the Japanese version leaned more towards Platonic Life Partners than romance. In contrast, the English dub pushed this to the point of writing a song about Misty's feelings for Ash. The latter is a point of controversy, and not merely among shippers - many instances of dialogue were rewritten in an attempt to push the couple, and even more notoriously, Misty's farewell sequence had the scenes depicted in the flashback completely changed (the original focused less on her as a Love Interest, and more on her as a trainer and as a friend).
This leads to a hilarious moment in the second season when one character asks Setsuna and Marina if they're in a relationship, which causes them to deny it simultaneously in a perfect, disinterested monotone.
Axis Powers Hetalia is a confusing case. While there are undoubtedly several potential pairings who get enough Ship Tease to justify calling it this, some characters are involved in more than one of these pairings, which leaves a situation where a character can have numerous other characters where a hook-up would have received enough subtext to not come as a surprise.
Sesshoumaru: 'Whether or not her death was in vain... is for me to decide.' (And whether or not his feelings were platonic or romantic... is for the fans to decide!)
While possibly not the case with their counterparts in Go Lion, Keith and Princess Allura in Voltron are widely considered the show's Official Couple, but their feelings for each other are never stated to go beyond friendship. It's possible that they are in love with each other and just don't realize it, but aside from a ton of Ship Tease the show doesn't really portray them as a couple.
Suzuna and Sena of Eyeshield 21. The two have a meet cute introduction to each other, people often think they're a couple, and Suzuna states that she doesn't view Sena as a little brother (unlike Mamori). Sena's feelings for Suzuna are left rather ambiguous as well, other then the fact that he's rather protective of her.
Roy and Riza are pretty much considered to be the Beta Couple in Fullmetal Alchemist. It's very clear they care very deeply for each other but it was never confirmed if they were in a relationship.
It's been confirmed through supplementary material that if they weren't both in the army, they would be dating.
In Megamind, Roxanne mentions that she was never in a relationship with Metroman, despite the public assumptions and the way he kept saving her from Megamind's schemes.
Live Action TV
Having arrived on and departed from the TARDIS at the same time, and already sharing something of a history, the Doctor Who fandom generally assumes this about Ian Chesterton and Barbara Wright. Ben Jackson and Polly, while not knowing each other as well before their travels, is also on the receiving end.
Video Games
The best example would have to be Princess Peach from Super Mario Bros., and it may extend to Princess Daisy from Super Mario Land. No romantic connection was made explicit, yet everyone assumes.
This is actually lampshaded by Luvbi in Super Paper Mario, and it's shown that the other characters seem to be just as clueless about whatever's going on between Mario and Peach, which leads Luvbi to think it is an unrequited crush.
The various games and other sources seem to go back and forth with how "official" Mario/Peach is. Some outright confirm their relationship, others only imply it.
The Paper Mario game before that seems to have NPC's throw out plenty of Ship Teases and the things Peach says while in captivity seem to imply that she thinks of Mario as a lot more than a friend, however the interaction with Luvbi is probably the only time where Mario and Peach's relationship is explored beyond the two knowing each other, unless you count Bowser accusing Mario of "trying to get between he and Peach."
Disgaea has a few of these, the most telling is the ambiguous relationship between Laharl and Flonne. While Laharl repeatedly claims to not like her, his actions in the game suggest otherwise, going as far as to attempt a Heroic Sacrifice. At the time, she was Not Quite Dead in the canon ending while in the normal ending he really does go through with it.
Persona 3 is quite heavy with two of the Love Interests, Yukari and Aigis. Yukari gets quite a few scenes with the Protagonist in the begining of the plot and Aigis is a full-blown Stalker with a Crush, justified that she sealed Death into the Protagonist but doesn't remember. Aigis is worse in that even when playing as a female in Persona 3 Portable she still gets subtext.
Rai/Kim from Xiaolin Showdown is a very popular ship on FF.net, and while there are indications that they might like each other in the series (Rai dreaming about being on a date with Kim, Rai being the only one of the guys that Kim kisses on the cheek), it is never outright stated that they like each other, and they aren't an Official Couple by the end of the show.
The third season episode "Dreamstalker" confirms Rai does indeed have feelings for Kimiko (making a thousand shippers go squee). However the Kimiko half has never been stated.
