Art Of Noise were intended at the very beginning to be a faceless non-group - in live performances, all the members tended to wear iconically-designed masks, and publicity shots were generally of things like spanners (with the ostensible reason that "A spanner is intrinsically more interesting than the lead singer of Tears for Fears". This was mainly at the insistence of the record company - the masks got dropped swiftly when the band jumped ship.
Daft Punk never show their faces. Their surealistic art-house film Electroma dwells on facelessness.
Guitarist Buckethead, who always wears a full-face carnival mask in public, and speaks through a hand puppet.
The band members of The Residents have never appeared publicly without their masks.
Lampshaded in recent years with the band revealing their "identities" as Randy, Chuck and Bob. (Carlos, the drummer, left the band to take care of his mother.) What this really means is new masks.
The members of KISS typically perform on stage in elaborate face makeup, and early in the band's history they refused to let themselves be photographed without it. Fans of the band speculated for years about what their idols really looked like.
Devo was a faceless band for a while in the mid-70s, performing in theatrical guises of Jungle Jim, The Clown, Chinaman and Booji Boy. Though fans can identify the current members, non-fans tend to go "The dude with the glasses"/"Mark Mothersbaugh" and the other four. Drawings of the band make this abundantly clear.
The band Lordi refuses to be photographed or seen without their elaborate monster costumes and makeup, in order to preserve their monstrous image. Even their real names are unknown to the public, with the exception of lead member Mr. Lordi. During the Eurovision contest they kept their costumes on constantly, with one shot showing the monsters peacefully lounging by the pool. A few Finnish tabloids have attempted to show Mr. Lordi without his costume and gotten a lot of criticism for it.
Averted with older masked metal bands, GWAR and Slipknot, whose identities are well known.
George Crumb's avant-garde composition Vox Balænæ is usually given the subtitle "For Three Masked Players".
MF DOOM always performs with an iron mask covering most of his face
Italian electro house musicians The Bloody Beetroots perform while wearing Venom masks.
Pink Floyd attempted to be this for much of their run. After their first album, they never showed themselves on album art, they refused to speak directly to the audience during performances, and they often used their famously elaborate stage effects to keep the audience from looking at their faces. This has mellowed somewhat in recent years, since they're frequently sought out for interviews now that the band has broken up. For a while, though, the vast majority of fans didn't even know the members' names.
Crimson Glory was one band that had full face masks, with the partial exception of the vocalist, whose mouth was open. However, as the masks were too hot to wear while performing, they got smaller and revealed more, and ultimately, they were photographed, filmed and performed live open-faced.
J-pop artist Kyary Pamyu Pamyu plays around with this trope very frequently in her music videos.
Japanese rockband Man With A Mission does this in spades. Each member constantly wears a wolf mask, as it's an integral part of their image.
There are no known images online of American rapper and producer TD Cruze. It is unknown whether this is intentional or not.