In The King and I, the King of Siam, as well as most of the other Siamese characters, speak broken English. (The show, as well as Anna and the King of Siam, the Margaret Landon novel and movie that inspired it, are banned in Thailand because of their historical inaccuracies and unflattering portrayals of the revered King Mongkut.)
Landon actually depicts the Kralahome (Prime Minister) as teasing Anna's son Louis by looking menacing and rumbling "You no can go!" She makes it clear that he's kidding; his English is much better than that.
The Indians in the stage musical of Peter Pan get this treatment. Their introductory song and dance has such gems as "Ugga wugga meatball!" Another one of their songs is called "Ugg-a-Wug," where most of the lyrics consist of "Ugg-a-wugg", "Gugg-a-bluck", "Puff-a-wuff", "Boop a doop," and so on. The Indians' spoken dialogue isn't any better.
Tiger Lily: We go up now. Keep guard. Watch for pirates.
Trouble In Tahiti has Dinah quote a couple lines of this from the "terrible, awful movie" she's seen (the stage direction calls for a South Pacific accent).
"Master Harold"... and the Boys has two African characters. One speaks in this form (albeit not as egregious as most of the other examples), the other speaks using proper English grammar. These are used to illustrate the relationships the characters have with the White Male Lead—Sam (proper grammar) is on equal intellectual footing, and approaches Hally as a friend, but Willie isn't and treats Hally as the master of the house.
Used by the native heroine of the 1923 play White Cargo (played by Hedy Lamarr in the movie). Her first line, "Me Tandelayo. Me good girl. Me stay," famously provoked the critic Robert Benchley to stand and announce, "Me Bobby. Me bad boy. Me go!" before stalking out of the theater.
Tituba talks like this in The Crucible. If you can't find a black actress to play the part, it's gonna be all kinds of jarring.
Billy in The Bat is a "Jap" who speaks in such phrases as "You give candle, please?"
William Shakespeareof course, with Princess Katharine of France in Henry V: "Your majesty shall mock at me; I cannot speak your England." She and her lady-in-waiting also have Funetik Aksents. The King and Queen of France, however, speak perfect English.
Miss Saigon: The entirety of the "Vietnamese" lines. Even if the gibberish written in the libretto can be deciphered, it doesn't make any grammatical sense. Amplified by the fact that actual Vietnamese-speaking actors are almost never to be found doing this musical...