In Yuru-Yuri, Nishigaki-sensei does this for Rise Matsumoto, who moves her mouth but can't be heard. Nishigaki-sensei will then tell the other characters what she said, or if they're by themselves will comment on whatever discussion they're having at the time.
In Baccano!, Claire Stanfield picks up the habit of speaking for his fiancee Chane, who is unable to talk. Being Claire, most people assume he's just making her responses up until she confirms that he does, in fact, know exactly what she's thinking.
There's an interesting take in Pirates of the Caribbean. The pirate Cotton had his tongue cut out, and has his parrot talk for him. The other pirates then interpret what the parrot says.
Mr. Gibbs:...so he trained the parrot to talk for him. No-one's yet figured how... Parrot: [squawk] Wind in the sails! Wind in the sails! Mr. Gibbs: Mostly, we figure, that means "yes."
Dogma: The Metatron speaks on behalf of God, since His voice is so powerful that human beings cannot hear it without their mind caving in and their heart exploding ("We went through five Adams before we figured that one out").
In Mystery Men, The Bowler does this for her father, the original Bowler, whose soul resides in a bowling ball.
George R. R. Martin's Haviland Tuf in "Guardians". He speaks for the mudpots of the planet Namor, who are secretly behind the planetary ecosystem's war against the human colonists (the colonists were eating the mudpots without realizing they were sapient).
Alternately parodied and played straight several times in various Discworld novels, especially those with Vimes. In Night Watch, Reg Shoe thinks he's speaking for the People, but Vimes reflects that the People would "clip [him] round the ear if they found [him] doing it". However, Vimes is quoted as using this line straight himself in the earlier book Feet of Clay ("Commander Vimes says someone has to speak for those who have no voices!"), and in The Wee Free Men, Tiffany Aching recalls her grandmother expressing a similar sentiment.
Although all but Matt consider Lanny to be The Voiceless, Matt repeatedly tells others that he finds it hard to get a word in edgewise when conversing with his friend.
Riva, a deaf mediator who appeared in the Star Trek: The Next Generation episode "Loud as a Whisper," traveled with a telepathic "chorus" that spoke for him. When they are killed in an assassination attempt, he has no way of communicating until Data builds a library to interpret his sign language, and the episode ends with Riva intending to teach his sign language to the warring parties. Why Riva did not simply write what he wanted to say was never addressed (the script implies that he and his family are illiterate).
Other times Howard will blatantly lie about what Raj said since he knows Raj can't contradict him out loud.
From the Mr. Bill segments of Saturday Night Live, Sluggo hated Mr. Bill, but Sluggo himself never seemed to speak, so Mr. Hands would tell Mr. Bill what Sluggo says, as well as carry out whatever harm Sluggo wanted to befall on Mr. Bill.
Rage Against The Machine have a song called "Voice of the Voiceless," which is about Mumia Abu-Jamal.
In the case of wrestlers who can't walk or have poor mic skills, it's very common to pair these wrestlers with a manager who cuts the promos on their behalf.
Guards at the Tomb of the Unknown Soldier, though mostly silent, can occasionally become this.
It goes without saying that for those who are mute or are incapable of communicating in a way other than sign language, they'll most likely have someone who can translate sign language.
In Shadowrun dragons can use a form of telepathy but can't speak the way humans can. They will choose a metahuman to translate their thoughts into speech when necessary, such as television appearances.
In Warhammer Fantasy Battles, the council of thirteen of the Skaven has the Grey Seers' seated member intercept their god's will; this really gives the Grey Seers a second vote in all matters.
Near the end of MOTHER 3, the recurring villain Fassad receives an interpreter robot after he's upgraded to some sort of half-mechanical nightmare out of the Book of Revelations. He can only communicate through horns.
There's a funny little side-story in Pokémon Mystery Dungeon: Explorers of Time/Darkness that culminates in someone speaking for an inanimate object. Dugtrio is obsessed with the VAAAAST SEA, often shirking work to go talk at it. One day, it answers back, telling Dugtrio that it really wishes they would get back to work... which, of course, turns out to be Loudred throwing his voice at the request of Dugtrio's son.
This is what Mario's partners are for him in the Paper Mario games especially in Thousand Year Door where there's tons of dialog between characters in the story, it's different depending on which partner you have out as it always reflects their personality but the basic story and their and Mario's current goal stay same.
Jonner is this for Mr. Blix in Freelancer while Trent is searching for von Klausen on New Berlin.
Arngeir in The Elder Scrolls V: Skyrim speaks for the other three Greybeards as their voice is powerful enough that if they try to speak to a normal person, they might kill them.
In Katawa Shoujo, Misha serves as an interpreter for the deaf-mute Shizune, who uses sign language. Some of Misha's own personality quirks occasionally get in the way, though.
In Captain SNES: The Game Masta, there are supposedly specially trained telepaths meant to give people suffering from Silent Protagonist Syndrome the ability to answer questions for game shows. Supposedly because the only one we see is the "Dodongo Dislike Smoke" old man, who speaks exclusively in his indecipherable lines from the game.
In A Miracle of Science, since the inhabitants of Mars are part of a collective consciousness, it is considered to be a Genius Loci, and thus every citizen speaks for the planet as a whole. This gets interesting once one of the Martians takes an interest in the main character.
Caprice: Mars likes you. Benjamin: That's like saying, "Brazil thinks you're cute."
Mars can speak directly if it wants to, it just usually doesn't because most outsiders find their Martian friend suddenly becoming a black-eyedecho-voiced avatar of the Hive Mind to be somewhat off-putting.
After being shot in the throat, Maine from Red vs. Blue can no longer speak, so Carolina gives him her AI Sigma to speak for him. Whether or not Sigma's words accurately reflect what Maine means to say is an excellent question.
Friend Bear speaks for Secret Bear in all appearances in the Care Bears. The only episode that Friend Bear doesn't appear alongside Secret Bear, Secret finds it most difficult to get the other bears to understand what he's saying.
Soundwave uses something of a variant. When he has to say something, he plays back a recording of something another character said earlier.
Deaf mermaid Gabriela in The Little Mermaid has her octopus friend Ollie translate her sign language.
Since B is silent, Dawn in Total Drama Revenge of the Island would speak on his behalf when he is communicating his plans to his team.