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Master and Commander: The French captain's face is never seen by the audience throughout the movie, until the end, when his body is seen lying in the sickbay. And then it's not even his real face yet, because the real captain swapped clothes with a dead marine, and dressed up instead as the ship's doctor, who pretended to be treating the 'captain's body' in the sickbay when Aubrey turned up (sort of a Chekhov's Gunman). The real ship's doctor turned out to have died months earlier. Of course, Aubrey and Maturin only find out once the other ship is out of reach again... which means, 'Beat to Quarters'.
Darth Vader in the original trilogy, although his face is finally revealed at the end of Return of the Jedi.
The Sand People of Tatooine cover their faces with goggles and wrappings to protect themselves from blowing sand. According to the Expanded Universe, they tend to get very offended if anyone tries to get a peek at what's underneath. The Dark Forces game franchise eventually revealed what the Sand People look like without their masks: vaguely like humanoid cats, with flat faces and drooping jowls.
The Jawas appear to be this as well. Very few of them in any material have ever removed their hoods—all anyone sees of their faces are two glowing red eyes. The Expanded Universe does mention that under their hoods they are "rodentlike" in appearance, and have "shrunken faces."
The Imperial Stormtroopers. We never see any with their helmets off, keeping them as faceless goons.
Boba Fett. The prequel trilogy revealed him to be Maori in appearance. In the Expanded Universe, the Mandalorian helmet is treated as being his "real" face, because it's what his identity is tied to. He occasionally takes advantage of this, because nobody will connect his human face to the legendary bounty hunter.
Darth Sidious, until Revenge of the Sith and later.
The Nazgūl from The Lord of the Rings. Their faces aren't hidden; rather they Nazgūl are completely invisible to the normal world, and Frodo can see them when he puts on the Ring at Weathertop (clearly in the book, distorted in the movie). If the Nazgūl take off their robes, living people see nothing; this is revealed in the end, where (in the book, and partially in the film) the head Nazgūl takes off his hood, showing a crown sitting atop an invisible head.
Octopussy's face is obscured when Kamal Kahn goes to see her early in the movie; only her hands are shown as she feeds her pet octopus while talking to Kahn. This, plus the way he takes orders from her and the fact that the movie is named after her made it seem like she would be the Big Bad and not him.
Flash Gordon (1980) - Klytus, Head of Ming the Merciless' Secret Police. His mask effectively obscures his features, though we're afforded a good view of his eyeballs and tongue when he gets tossed onto some spikes.
In The Dark Knight, Commissioner Gordon's daughter is at first completely unseen, and later we only see the back of her head (she looks to be about eight or nine). She is even only listed as "Gordon's Daughter" in the credits. Probably done to not get our hopes up about a possible future Batgirl appearance (or maybe just to not cause the confusion of the two Gordon kids both being Jrs. to their parents).
In the movie Ringu, future Stringy-Haired Ghost Girl Sadako is always shown with her long hair covering her face, even in flashbacks to when she was still alive. Her grisly, rotted features are revealed in Ringu 2, however, and her "good" half in Ring 0 Birthday always has her face on display.
Even in The Ring, the ghostly form of Samara remains with her face hidden behind her hair in almost all of her appearances. The Reveal of her face near the end of the movie easily qualifies as terrifying, if only for the expression on it.
Mara Chekhova, the operatic diva in Dario Argento's Opera.
In Raise the Red Lantern The Master's face is seen clearly only at a distance, and even that only a couple of times.
In Ernest Saves Christmas, Ernest has got a friend whose face we never see. All we see is his first person perspective, as Ernest clumsily bumbles about his house. This is a Shout-Out to Ernest's numerous commercials. Most of them had him talking about Product X to someone named "Vern", while "Vern" was doing something off-screen, like changing a lightbulb while Ernest held the ladder. Apparently, "Vern" is supposed to be a stand-in for the audience.
The Thing (1982) - John Carpenter invented another variant where the titular creature is seen frequently, but in forms which are obviously an ever-changing mishmash of other alien life forms, dogs, and/or men. Its natural shape is never revealed, even in death.
The 1976 film Mohammad Messenger Of God depicts early Islamic history. Mohammad never appears on-camera; his presence is indicated by the camera taking his POV, and an organ music leitmotif replaces his dialogue. It's longstanding Islamic tradition that Mohammad should never be seen in any form, under the premise that such depictions would run the risk of being used as idols, leading to worship of Mohammad rather than of Allah.
Treasure Planet - We viewers never get a good look at poor Jim's Disappeared Dad; some people on the Internet have taken advantage of this for video crossovers and/or fanart.
Irene Adler's enigmatic employer in the 2009 Sherlock Holmes movie ( A.K.A. Professor Moriarty) is presented like this in his appearances. We finally get to see his face in the second film.
In Citizen Kane, Thompson, the reporter whose investigation into "Rosebud" is the backbone of the plot, is only shown from behind or with his face in shadow. In fact, all of the reporters and news-media personnel in the film (those not employed by Kane himself) are The Faceless, reinforcing that what they do is more relevant to the story than who they are.
In Radio Flyer Mikey and Bobby's abusive stepfather "The King"'s face is not fully seen throughout the movie until the end, reportedly this was done because his actor Adam Baldwin didn't want to be associated with child abuse.
John, the mysterious alien Guide in Enki Bilal's Immortel ad Vitam has his head completely wrapped in black cloth at all times, presumably to protect himself from Earth's hostile atmosphere. When he finally succumbs to our air, he evaporates into nothingness, leaving only empty clothes behind.
An in-universe example actually happens to Mike Wazowski in the film Monsters, Inc., who for some reason, always has his entire face covered up in every single media he's appeared in, such as a commercial, a magazine, and even the DVD! (the hole in the middle of the DVD was actually placed in a way so that it too covered Mike's face). A more straight example would be a Godzillastand-in named Ted, who for some reason is seen only from the legs down and clucks like a chicken (he was originally going to sound exactly like Godzilla himself, but the idea was refused due to copyright reasons). A blooper at the end of the film actually revealed that his upper body actually belonged to Rex, here portrayed as a normal-sized Tyrannosaurus rex instead of a plastic toy.
In Let Me In, the English-language remake of Let the Right One In, a clever stylistic choice signifies how it is a film principally about children (more or less), where the adult characters are mostly peripheral and often fleeting. Owen's island-like status is emphasised by his absent father only making one scene, by telephone, and his mother a fairly constant presence in the book appears numerous times yet is never once seen properly on camera: she varies from being a distant figure, a ghostly reflection or obscured by a door, to fully visible yet thrown way out of focus or seen only from the neck down; even a passport-type photo glimpsed in her wallet is crumpled to the point of indistinguishability.
Averted in Judge Dredd. Although Dredd keeps his helmet on whenever he's on duty, he has it off most of the movie, unlike the original Comic Book.
In REC, the face of Pablo is never shown, given that he's the one behind the camera most of the time - and even when he isn't (after he dies), his face is never shown.
ET The Extra Terrestrial: With the exception of Elliot's mother, the audience doesn't see the faces of any adults until the final third of the movie, playing up the perspective from a child's POV.
In Julia's Eyes, during the middle of the film, in which Julia has her eyes bandaged, everyone she interacts with has their face obscured to the viewer; either we see them from behind or their head is cut off by the camera.
Gene Hackman's employer in The Conversation is only ever seen in shadows. Made all the more tantalizing because he's played by a very famous actor, who was unbilled in the film's original release, leading many viewers to say 'hey, is it really that guy?'
In the film adaptation of John le Carré's novel Tinker, Tailor, Soldier, Spy, the Soviet spymaster Karla's face is never shown, although we do hear his voice, see parts of his body and hear his name frequently as he is central to the plot.
Jigsaw in the Saw series. After the first movie, though, this turns into a strange example, as the identity of the original Jigsaw is known (one John Kramer) but as the series progresses "Jigsaw" becomes more of an idea than a single person.