Yes, it's perhaps borderline possible to publish, but keep in mind that your market will become tangible if it includes people who are intrigued by foreign cultures and interaction between them, but not specifically multilingual, or perhaps multilingual but not necessarily in your specific languages.
People are rarely as multilingual as their region. Although for example Switzerland has German, French and Italian as official languages, you will have a hard time finding people who speak more than 2 of those plus English. Of any of those, the portion of people interested in a novel of any particular genre will be tiny.
I have some grasp of perhaps half a dozen languages, but reading a novel which mixes them too liberally would be extremely tedious to me. I get severely confused when the language abruptly switches even just between the 3 languages I'm fluent in, it makes my head hurt, so even if you manage to rely on only the languages that I can read well, and though I have a deep interest in cultures and languages, I still probably wouldn't read your novel. If it's just showing off that you are good at multiple languages, it would come off as arrogant. I don't think I'm anyhow unique, I saw that happen at a hotel on Ibiza where the receptionist was fluent in 5 languages, but she had to talk to a guest who was switching between 4 of those every few words. When I walked up to her, she was staring into the distance, then at her papers, and into the distance again, and was thoroughly confused, and when I asked her, she uttered a disjointed bunch of words in all or none of those languages, asking for a few moments of silence to gather her spirits.
In 2010, "Julian Comstock: A Story of 22nd-Century America" was published and contains some untranslated foreign language and even does interesting things with the languages and the language barrier. Granted, it's written by Robert Wilson, an established science fiction author, and you will probably have a much harder time finding a publisher.
You must put yourself in the shoes of the publisher. Will the work still be enjoyable without in depth understanding of multiple languages?
Multilingual, or rather cosmopolitan literature has tradition. Countless classic Russian novels contain large swaths of French, because that day's reader was assumed to be fluent in French. Today however, they are printed with half page long footnotes in a terrible tiny font, making them extremely tedious to read for people who don't know French.
Perhaps you should reconsider whether novel is the right genre. Perhaps it should be a visual novel, allowing the reader to follow the story and infer the meaning even if they are not fluent in the language, by inserting visual clues, instead of increasing the verbosity of the text and making it feel superfluous to people who do understand the language. And it has to be compelling on a basic merit and not just because of multilingual puns.