In the Get Backers anime, whenever Raitei and the original Four Kings are shown in flashback, the 4th King, Kurusu Masaki is always hidden in shadow (Fridge Brilliance because he has the superhuman power to generate/manipulate light) and only his blackened silouhette is shown. This lasts until the first episode of the final story arc, where he is formally introduced
The Big Bad from Noein wears a spooky, warped golden mask that also muddles his voice and cloaks him in white energy. It turns out to be a good thing, because he's another quantum version of the male leads and is damn creepy-looking.
The parents and most of the adult relatives in general in Yuru Yuri. The audience usually just sees them from behind, and if you ever see their face, their eyes are usually obscured in some way.
In the Second Season of Tower of God, Baam's new self, Viole, has his eyes hidden by his blinding bangs.
The face of the villain Gargoyle from Nadia: The Secret of Blue Water was always hidden by a mask; said mask was finally destroyed near the end of the series. His actual face gets less than five seconds of screen time.
Soichiro Otonashi, Kyoko's late husband in Maison Ikkoku, has his face blacked out◊ in all flashbacks of him. Even his yearbook photo was obscured by a large ink stain. When the main character is finally able to see a picture of Soichiro, it's an Un Reveal for the reader.
GUN×SWORD: Shino is only ever seen once, and that is in silhouette behind a sheet as she hangs laundry in Ray's dying fantasy. However, we do see a waitress who Joshua says looks just like her. Also applies to Elena, who appears in several flashbacks; her full face is never revealed and we never see her eyes.
JoJo's Bizarre Adventure - The face of Dio Brando in the third series is never seen until a splash page reveal near the end and his Stand powers are also not shown until shortly thereafter. Although his face wasn't obscured in the slightest in the first volume, the art change since then leaves his true features a mystery, as his face is shown in shadow (often, the rest of his body will be perfectly visible). This is primarily because he has taken the body of the protagonist from Series 1, Jonathan Joestar. In the arcade and PSX fighting game, this trope is literally embodied in the character of Shadow Dio, whose face is blacked out and whose Stand is shown briefly as a colorless apparition from the waist up. His attacks do not reveal his Stand powers, and hardly use his stand at all, focusing on his vampiric powers instead. However, he can stop time just as the normal Dio can, has Dio's knife-throw as a standard move, and even has a special move where he stuns the opponent with his Stand, stops time, and surrounds them with knives as time starts again
In the same part, Midler, wielder of the High Priestess Stand, is never seen clearly. When she appeared in the aforementioned fighting game, they couldn't do this; therefore the game is the first time her actual face is seen. She appears to resemble a sort of belly dancer.
Given what Jotaro did to her stand a few minutes ago, there was definitely an excellent reason to not show her face.
Sabrac's head in Shakugan no Shana II consists solely of blue hair, wrapped bandages, and Glowing Eyes of Doom. Given that the villain before him was some sort of centipede energy cannon, he most likely doesn't have anything else. His body is actually under the city, apparently. They think they killed him but...
Revolutionary Girl Utena uses this trope extensively, in many different ways. Primarily there's the Shadow Girls, who are always putting on a shadow play and thus are only ever seen in shadow. All parents have their faces hidden as well, and everyone has their face hidden in flashbacks, even characters whose faces we normally see. Oddly, nothing obstructs the viewer from seeing their faces — their faces are simply not shown, appearing instead as a blank space.
Pokémon - Giovanni, the leader of Team Rocket, for a good part of the first season. In the first movie, when Mewtwo rebels on him, we get a decent look of him- albeit a Face Framed in Shadow. The anime fully reveals his face in a episode parallel to Mewtwo's escape and subsequent destruction of the base he kept the Pokemon in.
Many villains in One Piece. At one point, an entire team of villains wear masks, although it is explained that masks are just popular in that town.
In Trigun, the face of the Big Bad, Knives, stays hidden for most of the anime and the near-totality of the first manga. This is because the fact that he is Vash's twin brother is an important Reveal that comes fairly late in both the anime and the early parts of the manga. At the end of the first manga, his face is still hidden behind huge clunks of wavy hair and at the beginning of Trigun Maximum, he wears a mask for a few volumes before his face is finally drawn in detail around volume 4.
In InuYasha, Onigumo, the human core of Naraku, is never shown, and his face is Unrevealed often. When Kikyo was taking care of him, his face (and much of the rest of him) was covered in bandages. When we first met Naraku, he always wore his baboon pelt outfit, with face never seen within the hood. He took on a disguise early on (some young feudal lord), and it's the disguise's face that we start seeing under the hood on occasion. Eventually, he discards the baboon pelt when it outlives its usefulness, but again, the face we've come to know as his is simply that of the feudal lord from his first full appearance. When Naraku separated the Onigumo-ness from him by turning it into another, humanoid demon, the demon had a featureless face, and took a face (and a name) from a monk who tried to vanquish him. In flashbacks with Onigumo's cohort Rasetsu, Onigumo's face is always kept in shadow.
Neon Genesis Evangelion thrives on this device. Yui's facial features are a mystery for most of the series to hide Rei's striking resemblance to her and to emphasize Yui's value as an archetype. The face of Asuka's mother, Kyoko, is only shown once, in The Movie (and even in this case, it's not very clear), and her father's and adopted mother's are never shown. Kouzou Fuyutsuki's students in episode 21 are also The Faceless, strangely enough. As for Shinji's caregivers in the manga, they are The Eyeless! This device seems to be partly meant to accentuate the theme of alienation in post-industrial Japan... oh forget it.
For fairly obvious reasons, the face of the pilot of Eva-03 is hidden until the end of episode 18 too, though the character's identity is strongly implied throughout episodes 17 and 18. Also note that both in the manga and in the official spin-off (!) Neon Genesis Evangelion: Shinji's Complementarity Project, the kids and adults who bully Shinji and Rei, respectively are also The Faceless, which adds to the creepiness factor and might also emphasize the mindlessness of bullying and/or the interchangeability of mindless bullies.
And in the yet-another-official-spin-off Neon Genesis Evangelion Gakuen Datenroku (Neon Genesis Evangelion Academy), Gendou's face is hidden till the very last page of the first chapter in which he appears. Probably because he's supposed to be missing, but we know what he looks like and it's not like we didn't expect it (Shocking news! Gendou Ikari is the head of the Nerv! And he might be scheming something involving teenagers and super-weapons!!). Also, so far Shigeru Aoba, Makoto Hyuga and Maya Ibuki (the operators of the main series) are only seen from behind if at all. Many of the Angels are completely hidden in darkness when they appear.
In Tsubasa Reservoir Chronicle, the real Syaoran's parents' faces' are always hidden or shadowed when seen, his mother getting more shaded out than his father, for spoileriffic reasons.
Kakashi wears a ninja mask over the lower half of his face all the time except when he eats - we never see his full face, fan edits aside. Shino Aburame averts this trope, though, because while his face is usually concealed from the nose down by his collar, it is revealed from certain angles during fights. He always wears his dark glasses, though.
One episode has Team 7 trying to see his Kakashi's face without the mask. At the end of the episode he shows them, but cops out with another mask underneath.
One chapter's cover shows him in bed maskless but has him covering his face with his cover and hand.
Likewise, the cover for chapter 245 has him without his mask, with part of his nose and face showing, but his mouth is covered by his hand.
Tobi's face was consistently hidden by shadows or a mask for a long time. This inspired a great deal of speculation among fans as to who he was. His identity was complicated even further by the fact that he isn't the person he claims to be.
Pain, the leader of Akatsuki, was this for a long time, appearing only as a silhouette. He eventually does show himself. The concealment was probably meant to hide that his appearance changes because he has multiple bodies.
Clamp School Detectives has the weirdest example yet: the school Chairwoman. Not only are her eyes either shaded out or off the screen, but she often hides them purposefully with either a fan or a set of blinds for no apparent reason other than that the camera could have gotten a good shot of her face.
Manga example: The Demon Ororon has God, with a massive set of emo bangs that hide his eyes entirely.
Two examples from Dr Slump: one is a police officer never seen without his Stormtrooper helmet, the other one is a guy appearing in the last storyline of the manga, a biker who can't get off his bike due to some illness. He never ever removes his helmet, and not even his name is ever given.
Zolf in Slayers goes from some sort of inhuman mummy man to a guy that could best be described as looking like Link's uncle from The Legend of Zelda: A Link to the Past. Suddenly he's a lot more human and respectable of a character.
Beyblade - Although seemingly a harmless computer geek, Kenny obviously had something to hide. Not only were his eyes always covered by his hair (they were shown once, in a Face Fault), in the original version he was also known almost exclusively by a nickname. Suspicious.
Ghost in the Shell: Stand Alone Complex has the Laughing Man, who is such a competent hacker that he censors out his own face with a logo not only from any simple cameras that see him, but from the computerized brains of anyone who looks at him, in real time. Of course, he is revealed to viewers around halfway through, and he also shows himself to Section 9 after the main conflict is resolved.
A vast quantity of Doujin mangas involve female characters being sexually involved with male characters whose faces are either obscured, drawn with scant detail, or in extreme instances with no face at all. The reasoning is most likely because the artist believes that the reader would prefer to fantasize about themselves being the guy doing the filthy things to said female character, rather than watching one of the official characters or some some stock thug/nerd/old man/etc do it. For the same reason, many H-games have the same black faced or eyeless wonder.
Makoto Numata in Kurosagi Corpse Delivery Service wears shades all the time. So far, his eyes have been visible only once, through his shades, for the sake of a reaction panel.
Morganna Mode Gone of the .hack series takes this even further: she has no physical body. That's because she is The World; she is the game and everything in it, given an intelligence. The only "appearance" she has is either a disembodied voice or the Phases, programs created to do her bidding.
Hayate's parents in Hayate the Combat Butler have kanjis for their faces. His brother's face is darkened/obscured in his appearance up until chapter 263.
In Fist of the North Star, the Last General of the South Star is always shown wearing a helmet until it is is revealed that the Last General is none other than Yuria, Kenshiro's thought-to-be-dead fiancee. An antagonist named Kaioh is later introduced whose face is initially concealed by a helmet when he first appears. This is done to hide the fact that he is Raoh's blood-related elder brother (which was already obvious, considering how similar he is to Raoh in every other aspect and the fact that they pull the same plot twist with Kenshiro and Hyoh).
Tokita Mako in Mx0. Her facelessness is part of the joke that her chest is so big that anyone can recognize her with her chest alone. Her face is revealed in the final chapter.
In Monster, this was Johan's first appearance as an adult. His face is not shown until he shoots Junkers, having first been obscured by shadows and then through the back view. At the end of the exchange, both Dr. Tenma and the audience get to see his face. When he's shown to be a Bishōnen young man smiling pleasantly, it's even creepier.
Sailor Moon has Professor Tomoe, Sailor Saturn's crazy father. For a long time, all you see is a black shape for his face, a strange red smile and bright shiny glasses.
Kodocha has Rei, protaganist Sana's "pimp" (actually her talent agent; Sana likes to exaggerate) who is constantly wearing sunglasses, even in the shower or in bed. Sana begins to wonder what he looks like without them, but he refuses to remove them. Toward the end of the first season, he finally takes them off. He then turns his perfectly normal face to the viewer and says, "sorry I'm not deformed or anything."
Eyeshield 21 uses this trope to build up tension for the final game of the series, a showdown between the US and Japanese football teams. The American team's most feared player, Mr. Don, is shown only from behind for several chapters, and even a group shot of himself with his teammates has him facing away from the camera. This serves two purposes: it makes it more dramatic when he finally takes the field in the final game and we see his face, and it hides the fact that his father is the president of the United States.
Likewise with Hiruma's father, who the audience never sees his face. It's either shadowed or he is shown from an angle that blocks his face.
It is likey that, given what the son is like, Hiruma's father is probably the actual Devil himself.
In MÄR the Chess wears masks for the most part generally taking them off for good once they're revealed. Ash takes his off to at the end of his battle, to show his face in the same expression as his mask, but replaces it. And he's the one who loves kids. Halloween's mask is his face. He's actually a childhood friend of Allen.
In Angel Sanctuary, though only in the manga since the anime didn't get that far, the head of the Council of Angels Sevothtarte hides his face behind a mask. It's said to hide uncountable scars he received from the previous war, but it serves another purpose. To hide the fact that "he" is a woman, and her mark as a Fallen Angel.
The Teacher from The Sky Crawlers remains unseen throughout the entire feature; you only see his plane in action.
In the Higurashi no Naku Koro ni anime, you never see the parent's faces, for some reason, just a voice and maybe (if you're lucky), part of their body. However, they're shown fully in the manga.
The rank-and-file Hollows of Bleach can be considered this. From the moment they are created, a bone mask covers their face. This allows for them to largely be dehumanized by both the Shinigami and the author despite their often tragic origins. A major reason that Shinigami are taught to attack them from behind is so that they won't see their human faces revealed in their death throes.
The Arrancar, having removed their masks, are given a considerable amount of characterization and their deaths are not as easily brushed aside as those of common Hollows.
The Conductor of Galaxy Express 999 looks rather like a Final Fantasy Black Mage in a peaked cap. On one occasion, the cap fell off and he had no head; later, an assassin who went after him in the shower was deeply traumatized by his true form.
Urhi Shalma's face is hidden by a cloak for several chapters, until it's revealed later on. Even later, his face gets hidden again via a cloak when he's presumed dead but is revealed once again.
The Brunette from Ben-To to the point of being a running gag where even the official character website's photo of her doesn't reveal her face.
Glen Baskerville from Pandora Hearts. So much, in fact, that in Lotti's memories, shadows obscure not only his face, but the top of his red coat as well.
From Kuroko No Basuke, we have the former captain of the Generation of Miracles, Akashi. He's always shown from the back or the neck down. Even when he's face-to-face with the main characters, he has a Face Framed in Shadow. His face is finally revealed at the end of chapter 113.
This has been parodied on the skit High School Boys and Older Brothers and Sisters, where Yoshitake didn't even know what his older sister looks like.
Yattodetaman, one of Tatsunoko Production's Time Bokan series, has as recurring characters a newscaster and his cameraman who commented upon the mecha battles between good and bad guys. The cameraman never spoke and was so tall that his face was always out of the picture frame. Sometimes he was so tall that could be seen only from the waist down!
In Medaka Box, Hanten Shiranui almost always faces away from the camera. The few times he's facing front have his head obscured by something (usually a speech balloon).
In Gundam SEED, Big Bad and resident Char Clone Rau le Crueset's face is never shown during the TV series, but the Compilation Movie version of the plot does. Except briefly when it get's blown off by a gun (and partially covered after by Rau's hand) in episode 45 (43 in the remaster) and in flashbacks in the same episode where we see him meeting Mu as a kid.
In Gakuen Babysitters, Yutaka, Midori's father, usually appears with an overgrown full beard and a hat, obscuring his eyes and the shape of his face. When he "appears" clean-shaven in the Volume 5 extras, he's only seen from behind, and when Midori cries due to not being able to recognize him, her mother is forced to cover his still unseen face with a Santa mask to get her to stop.
In Saki, until the fourth round of the prefectural finals, Momo's face was never shown from the front, and she had no speaking lines. At the end of the first East-South round, however, she surprised everyone by calling a tile Touka had discarded, with her face being shown for the first time. Afterward, a flashback sequence ensued, showing that for much of Momo's life, she was ignored due to lacking presence.