Plautus, as it happens, uses Punic (the Phoenician dialect spoken in Carthage) for some extremely bad bilingual puns in Act V, Scene 2 of his Patruus aka Poenulus. Agorastacles (born in Carthage but raised elsewhere) and his slave Milphio have just run into Hanno, an older obviously-Carthaginian man, and they try to communicate with him in Punic.
In the following, anything in bold is Punic, anything not in bold is Latin, and I've left out some lines for brevity:
M: Should I talk to him in Punic?
A: You know Punic?
M: No Punic can be more Punic than me, nowadays!
A: Okay, go ask what he wants, why he's come here, who he is, where he's from, what country…don't hold back on the questions!
M, turning to address the stranger: Um, yes. Hello!
H: Hello, sir. (Avo, donni.)
M: He, uh…wants to give you something! He wants to give you, um, I don't know what…but you can hear it, right? (Latin donī, "gifts")
A: Good. Greet him in Punic for me?
M: Hello, sir, he says, those are his words.
H: Is he a friend of yours? (Mechar, bocca?)
M: Uh, better you than me, I guess? He's saying that his mouth hurts. Maybe he thinks we're doctors? (Latin bucca, "mouth")
H: There is no doctor here, you absolute idiot.
A: Ask if he needs something?
M: Hey, you, the one with the silly clothes! Why did you come to this city? What are you looking for?
H: Excuse me?!
A: What's he saying?
H: Who's going to answer the stranger? (Mi uulech ianna?)
A: Why did he come here?
M: What, are you not hearing him? He says he wants to donate some African mice for the parade before the aediles' games. (Latin mus Africāna, "African mouse")
H: Go screw yourself. (Lech lachannani limini chot.)
M: He says he's imported, um…spoons, little pipes…and nuts! (Latin ligulas, canalis meī, nuces, "my spoons, little pipes, nuts")
H: He's lying. (Assam.)
M: Yes, roasted nuts, roasted in oil. (Latin assam, "roasted [nut]")
H: You are an imbecile. (Palu umer gad etha.)
M: He says he's been given…shovels…and pitchforks…to sell? I think, unless you have a better idea. (Latin pāla "shovel", mergās "pitchforks")
H, realizing something: These people aren't actually Phoenician… (Mu Phonnim si corathi…)
M: Oh, look out! Don't do what he said just now!
A: …what did he just say, or ask? Explain.
M: That you should order someone to put him under a basket, and then put a whole lot of stones on top, in order to kill him! (Latin pōnī sub cratim, "put him under a basket" (*))
H: By the grace of the Lord of Heaven, he needs to shut up.
M: …yeah, I have no idea about that one.
H: For your sake, I'm going to start speaking Latin now.
(The Latin translations are my own; the Punic translations are based on Wolfgang de Milo's appendix to LCL 260.)