Calypso in Pirates of the Caribbean: At World's End, when in 50 foot giant form, rains unintelligible things at the pirates after they set her free. Word of God says that she's cursing them, and basically telling them to go fuck themselves.
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.
C-3PO is a "protocol droid" designed to help with interspecies communication. This means it's his directive to translate between various languages (of which he knows a lot), and he does it quite often.
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.
After which of course, Mumbles is able to speak coherently to confess Big Boy Capri's plans. Albeit, he's still mumbling slightly and speaking kind of fast.
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.