Tsubasa Reservoir Chronicle plays with this. In the beginning Syaoran and Sakura already too close but have yet to tell each other their feelings. However, Sakura's amnesia causes the relationship to reboot with Syaoran also distancing himself due to the fact that in order to help to her memories, he pays with all of her memories that include him. However, across the series the two start feeling attracted once again until the two nearly die and confess their feelings.
Rurouni Kenshin went through this so long, it almost got annoying. Kaoru was too shy to tell Kenshin how she felt, and Kenshin wasn't sure if he deserved someone like her, especially since he accidentally killed his first wife Tomoe.
Rito and Haruna in To Love-Ru are the very definition of this. Rito tries confessing a few times but is always interrupted. She takes so long that while Rito still loves her, he also falls in love with Lala, making the situation even more complicated.
Kiri and Elraine from Double Arts. She's too shy and thinks he's clueless about her feelings, and he's convinced that the boost in sexual tension caused by confessing to her would make life trapped together too awkward.
Junpei Manaka and Aya Toujo in Ichigo 100%. Actually, this trope is even the main reason why this series DOESN'T end with a First Girl Wins scenario.
Kind of exists between Kenichi and Miu of Kenichi: The Mightiest Disciple. It's slightly subverted in that they're very good friends, but Kenichi is too nervous to take it to a romantic level, and Miu is incredibly jealous of any other girls who hang around with Kenichi, but she doesn't take the initiative either. Also a case of Everyone Can See It, as all of the masters are totally aware of it, and are watching to see what will happen.
Later on in the series, this trope is subverted in a unique way. Both Kenichi and Miu acknowledge their feelings for each other (well before the end of the series at this point by a 100+ chapters mind you) but they decide not to date until Kenichi has enough power/skill to protect Miu. In the last page of the entire series, a small picture is shown on the side of the page that depicts a small girl is shown between two unknown figures, implying that not only Miu and Kenichi started dating, but eventually married.
In Full Metal Panic!, Sōsuke and Kaname fit this trope to the "T" — especially later on. Kaname has problems taking on the aggressive role of romantically pursuing Sōsuke (as she's shown to constantly be waiting for Sōsuke to take the initiative, and is constantly disappointed when she finds out she misunderstood his actions as being romantic advances). And Sōsuke... has hugeissues with romantic intimacy. He's shown to become very nervous when confronted with anything "inappropriate" and too sexual, and has a tendency to clam up or run away. The most he appears to be able to muster with Kaname is to shyly hold her hand, which results in some highly frustrating Will They or Won't They?.
Examine the freak-out he had in the beautician shop (he didn't at all like the idea of someone holding blades near him.) Then examine how, with Kaname holding a pair of scissors near his head, he is somehow relaxed enough that he falls asleep while she's trimming his hair. He's not even that relaxed around Tessa, or Kalinin. They Will. And They Did.
Sawako and Kazehaya of Kimi ni Todoke. She's shy, innocent and unassuming to a fault, he's a truly genuine Nice Guy who doesn't want to push it. Without Yano, Chizu and Ryu nudging them, they would probably both graduate before they got around to doing so much as hold hands.
Ran and Shinichi of Detective Conan. Every single time Shinichi turns back, they just keep beating around the bush instead of telling each other how they feel. For 16 years (not in story). Straight. Shinichi did end up giving Ran an indirect confession in the London Arc, though.
In Bakuman。, Mashiro's uncle and the woman of his dreams Azuki's mom were completely crippled by this. It's likely that Mashiro and Azuki would have fared even worse, if Takagi hadn't dragged Mashiro bodily into speaking with her and agreeing to draw manga.
I"s would probably be about half the length if it weren't for this trope.
Feldt and Setsuna in the Gundam 00 movie show signs of this. Feldt's feelings for him are so obvious Everyone Can See It but Setsuna has his own intimacyissues, not to mention his recent transformation into an Innovator. The farthest they get is a hug when Setsuna wakes up from his ELS Mind Rape-induced coma... at least that's what we see; exactly what happened between the hug and Setsuna calling the bridge that he's taking the 00 Quan[T] (with a teary Feldt following closely) is left up for interpretation.
The main problem is that after entering the final battle, Setsuna goes to the ELS homeworld and they never meet again. Even if they were to meet after the epilogue, Feldt is already in her seventies while Setsuna is most likelyimmortal.
In Nyan Koi!, both Junpei and Kaede like each other, but neither one seems willing to make the first move. Other girls expressing interest in him also serves to confuse Kaede as to whether he's dating them, or may not really be interested in her.
In Papa no Iukoto o Kikinasai!, both Yuuta and Raike are interested in each other, but neither one seems able to admit their feelings to the other.
Raku Ichijo and Kosaki Onodera of Nisekoi quite blatantly have feelings for each other, however they are unable to admit their feelings for each other. Of course, the fact that Raku has to pretend to be in a relationship with another girl due to family issues doesn't help.
In Kase-san, both girls get plenty of hints from the other that they're interested, but both are way too awkward and nervous to do anything about it for a full year in-universe time.
Peter Parker with many of his early love interests. He just thought no one could be interested in a dweeby little science nerd. Once he hit off with Gwen Stacy, the rest was history.
Cloak & Dagger: Ty's too shy to ever think a girl like Tandy would be interested in him. Tandy thinks Ty only loves her as a friend. Cue the UST...
In a FoxTrot arc, Paige and a boy in her class were in this situation, but overcame it and went to a school dance together. After this arc, the boy disappeared and Paige went back to her usual state of being unable to get a date.
Scott and Jean in the early days of the X-Men could very well serve as the page image.
Naruto and Hinata in Escape from the Hokage's Hat. Hinata's a Shrinking Violet with low self-esteem who still thinks he holds a torch for Sakura. Naruto doesn't want to jeopardize one of the only healthy female relationship he has and is afraid to be rejected. It's takes a Freudian Slip to actually help them.
Naruto: "If I was to ever date a girl, I hope she's as cute as you are." (beat) Oh Crap face then Facepalm
In Tomorrow's Doom, it's quite obvious that Amaya and Tadao like each other, but neither wants to admit their feelings to the other. Aiko finally decides that enough is enough and nudges the pair toward each other during the festival.
The Stalking Zuko Series, it becomes obvious to everyone that Katara and Zuko both like each other due to their history and self-esteem issues, they think they are not good enough for each other or are in love with someone else.
Lance and Gwen's relationship in The Student Prince. Neither can work up the courage to ask the other out for a long time, both of them thinking the other is out of their league.
The Dish: Glenn and Janine have this in spades. Drives others mental.
In Invisible Stanley, one of the sequels to Flat Stanley, Stanley helps a shy couple get married by creeping up invisibly behind the man, imitating his voice, and proposing to his girlfriend.
Nick and Lirael had the subtlest of hints towards a mutual infatuation in Abhorsen, which were complicated by his death. Luckily he got better, or else Garth Nix wouldn't have been able to confirm their infatuation in the short story "Nicholas Sayre and the Creature in the Case".
Piro and Ibronka in The Viscount of Adrilankha.
Ibronka has got one of the best lines in all of Dragaera on the subject: "My dear Viscount, come over here and kiss me before I die of embarrassment"
Edward Ferrars and Elinor Dashwood in Sense and Sensibility. Both are calm people who do not crush easily and hide their feelings. Proper Elinor is aware that she must not appear too eager, especially when his family strongly disapproves of the match. Moreover, Edward is the Master of the Mixed Message and involved with somebody else.
Beta Couple Jane Bennet and Mr Bingley in Pride and Prejudice. Jane is usually sweet but composed, so it's obvious only to her family and close friends that she likes Mr Bingley. Mr. Bingley thought she liked him but was easily persuaded by his friends that she was indifferent. Had there been no meddling, those two would have got egaged after the Netherfield ball in the third of the book.
Manny and Roweena in Black Books. It takes Bernard bellowing "Offer her something, you fool!" at him to get Manny to make any headway with her.
Jeff and Julia in Coupling. (A rare example where the new character actually stuck around, as opposed to mysteriously vanishing by the next episode.)
Kaylee and Simon in Firefly. Although in Kaylee's case, it's less that she's shy (hell, she got her job on the ship thanks to a one-night stand with the previous mechanic), and more that she feels inferior around Simon.
The page quote is from season 1, where Robin was pining over Ted who had been pining over Robin, but had just gotten over it and moved on.
Barney and Robin in the fourth season. Funny when you consider how much of a Handsome Lech Barney is. It seems like that old stuff: quick flings are one thing, but getting into a steady relationship is something completely different.
One episode of Sabrina the Teenage Witch had Valerie be too shy to ask out a guy. Sabrina decides to intervene by calling Cupid and asking him to make the guy in love with Valerie, but Cupid responds it would be a waste of an arrow because it's already clear that they both like each other.
"I've Got This Friend" by The Civil Wars, with the obvious implication that the "friend" each singer is talking about is actually themselves, while the other suggests that their "friend" sounds perfect for them.
Anton Chekhov's play The Cherry Orchard. Self-Made Man Yermolai Alekseevich Lopakhin, the buyer of the titular orchard, has a troubled relationship with Varvara aka Varya, the Tsundere adopted daughter of its former owner, Liubov Andreevna Ranyeshkaya. In one of the last scenes, he works up the courage to ask Varya to marry him, but ultimately feels too shy and guilty to do so. This scene fits in well with the overall tone of the play —which has all of its characters suffering because of their inability to act decisively.
In the musical Double Digits, the duet "Oh, no!" is sung by a popular jock-boy and a quiter girl who is tutoring him in maths. To whit:
Sora and Kairi from the Kingdom Hearts series. At least in the first game.
The Legend of Zelda: The Wind Waker: In a subquest in Windfall Island, you can photograph two persons who have this problem. (They share a look.) The quest-giver will then use this photograph to help them along.
In Skyward Sword, this is the situation between Pipit and Karane, Link's fellow students at the Knight Academy. When Karane receives a love letter from another student, Cawlin, her public rejection of Cawlin's affections leads to her and Pipit revealing their feelings for each other.
Some Fallout fans speculate that this trope is going on between the Lone Wanderer and Amata in Fallout 3.
A cut ending to Knights of the Old Republic 2 would have a female Exile's relationship with Atton resolved to look like this was going on for almost the entire game.
Used as a running joke in Mother 3 between a woman and man, both with almost identical dialogue. Their relationship progresses through the game.
A running theme in Immortal Souls for John, who's too worried about others finding out about his secret life to seriously flirt, and Allison, who worries about whether or not he's doing OK too much to ever get a chance to get beyond that. Though Allison finally eventually perks up enough courage to ask him out for coffee by the end of the first game.
Elan and Haley from The Order of the Stick are this for a while, which can be best seen in their Will They or Won't They? plot near the beginning of the Azure City arc. Due to character personalities, it's played out as Elan being Oblivious to Love and Haley deliberately refusing to ask Elan out because she is taught to mistrust people. But as it is seen in this strip they are both attracted to each other for a very long time.
The movie more or less confirms this. Right before they have their memories wiped, Isabella kisses Phineas. Phineas is both very surprised and very happy.
Candace and Jeremy seem to be a canon example, though for a long time it was hard to tell if Jeremy even liked her back. Season two made it a bit more obvious that he was nervous around her, with the episode "Nerdy Dancing" specifically switching their usual dynamic (he was trying to impress her rather than the other way around).
Bright Eyes and Lancer in My Little Pony Tales until their friends' failed makeover attempts forced them to come clean and admit their feelings for each other in a Be Yourself moment.
On Goof Troop, PJ got a crush on a girl in his class, Rose Deckenbloom, after she recited a poem about being a Shrinking Violet, which he could relate to and thought was beautiful while everyone else in the class hated it. Then he spends much of the episode following everybody's bad advice about how to get her to like him. He tries to give her a present, but is too shy, so he asks Max to do it for him, which he does. Rose mistakenly thinks Max likes her and is delighted, but once they are at the dance she and PJ start to hit it off at the punch bowl... until Pete comes in and "helps" (he was trying this time, honest!) which causes Rose to run away in tears, thinking PJ doesn't like her. He then recites the opening lines of her poem and confesses to all the things he tried, she says she likes his true self more, and then this happens: