Isle of Dogs has Atari speaking in only Japanese, apparently without subtitles, and while only his fellow Japanese can understand what he's saying, the dogs can't. Lampshaded by one of the dogs named Duke, who mutters, "I wish someone spoke his language."
Many aliens and droids in Star Wars are unintelligible, speaking only in beeps, growls or alien speech. R2-D2 and Chewbacca are both examples. Almost everyone can still understand each other, leading to a lot of Bilingual Dialogue. However, there are occasional instances where characters will translate for each other. Examples include the two cantina ruffians and C-3PO's stint as a translator for Jabba the Hutt.
The Death Troopers in Rogue One are this to the audience. Unlike regular stormtroopers that are perfectly understandable even through their helmets, the Death Troopers sound like garbled radio static whenever they talk. Justified though, as it's supposed to be some sort of vocal encryption their higher-end gear provides.
Captain Stimpson in Texas Across the River is capable of speaking normally, but whenever he issues a command, it sounds like, "Haroo, har!". His soldiers invariably follow the command, despite themselves being unable to understand what he's saying.
Mikey O'Neil in Snatch. speaks with such a thick "Pikey" accent that outsiders have difficulty understanding him. His subtitles occasionally revert to "??????"
Mumbles in Dick Tracy, played by Dustin Hoffman. A stenographer is seen giving up entirely on translating. Tracy is able to discern what he's saying by recording him, and then playing back the tape slowly. Later, Mumbles is able to speak coherently to confess Big Boy Capri's plans.
Police Constable Bob Walker of the Sandford Police Service in Hot Fuzz speaks with such a thick rural accent that Nicolas Angel needs other villagers to translate for him. However, isolated country folks are unintelligible to everyone but Bob, forcing a translation chain. Interestingly, as Angel gets more familiar with the town, Bob's dialogue becomes more intelligible.
Fenster in The Usual Suspects. Benicio Del Toro invented this character aspect because the character was too thin, and existed only to die. He based his performance on Dustin Hoffman's performance as Mumbles. Cast members were asked to have him repeat himself if they couldn't understand his line delivery. A few of these end up in the final cut.
Farmer Fran in The Waterboy has a completely unintelligible Cajun accent.
Frenzy in Transformers live-action film speaks almost entirely in Cybertronian. However, he does speak English sometimes, such as his last words: "Oh, shit." Shockwave from Dark of the Moon is also fairly hard to understand.
Blaine Cody from Rat Race, due to a self-made tongue piercing. His brother, Duane Cody will sometimes "translate" his speech.
The eponymous Pootie Tang. It is said that he is too cool for words and nobody undestands what he says, but what he means.
Herbie The Love Bug embodies this in that his horn is his only method of communication. At least one person from each movie understands him.
Played with in Enchanted as far as understanding The Unintelligible goes. When Pip enters the human world, he finds himself in the difficult position of only being able to utter word-like squeaks. Prince Edward thinks he understands him perfectly. He doesn't.
Elmo in Schizopolis speaks in a weird, drawling language that includes such expressions as "Arsenal! Nose army," (a greeting), "throbbing dust expression" (roughly, "nice place"), and "ambassador landmine" (roughly, "let us copulate"). When compelled to speak in normal English, he becomes almost psychotically frantic.
The titular protagonist in Spider is a paranoid schizophrenic who mumbles his words incoherently much of the time.
Used for a fairly complex gag in Hot Fuzz: Farmer Webley has a combination of a severe mumble and thick west country accent that requires the local police to bring around a slightly less mumbly officer to translate for him, who in turn needs a more articulate officer to make sense to the cop from the big city.