A closer translation would be "let"; for instance "lasciami stare" (let me be) or "lascialo andare" (let him go). In this case it would be "you cannot let me become sad", but the intended meaning might indeed be not allowing me to become sad. The online dictionary wordreference.com does list allow together with let.
Let's put it like this: the Italian definition of "lasciare" (http://www.treccani.it/vocabolario/lasciare/) has a pretty long number of meanings; among those 6.a: "Seguito da un infinito (o da che e il cong.), è per lo più sinon. di permettere" (followed by infinitive, or "che" and subjunctive, it's mostly synonymous with allow). 6.b has a number of locutions where "let" is more appropriate, as in "l. andare qualcuno, non trattenerlo, concedergli di partire" (letting someone go, not stopping him, allowing him to leave).
Thanks as always for the clarification BUT I still wonder how helpful it is in language-learning to use a popular verb like "lasciare" that in popular usage means"to leave" in ways that are four and five and six definitions down the dictionary definitions. I'm trying to suggest that duolingo -- again a program I like immensely -- could think about changing "Non puoi lasciare che io diventi trieste" to something that captures the sense of the sentence in a simpler way. I'm thinking the English sentence may need adjustment since this sentence/translation is in the present subjective section, as I recall, and the subjunctive in English, to my mind, is a different beast.
Luckily we have the same use of "lasciare" in German ("lassen"). The basic meaning of the verb is: passiveness of the subject, i.e. in the case of this sentence here: You cannot stay passive while I become sad. "I let my children drink alcohol" (allow): My children drink alcohol and I stay passive. Italian "lasciare" = to leave a person: I go away and leave the other person behind (do not actively take her with me). And so on ... all can be explained in terms of the non-activity of the subject towards the object.
You nailed that brilliantly where it comes to the German translation - lassen/zulassen, passiveness is the lever to understand the meaning of lasciare in this context. But I still can't find the word for an equal translation of lasciare to English. The German lassen doesn't translate without changing the sentence by adding other words and allow, tolerate or permit have their own translations to Italian. So what would be the equivalent to zulassen in English?
My verb book has lasciare che as a separate section. to let/allow "lascia che pianga" let him cry and "abbiamo lasciato che Margherita stesse alla festa fimo a mezzanotte" we allowed Margherita to stay at the party until midnight.
And lasciarsi, to let oneself be, leave each other, say goodbye, split up
Well, being a beginner, I probably have it all wrong, but the given translation is (Google translate) 'Non puoi lasciarmi diventare triste'. That being said, being bilingual French and English, I see this as 'You cannot leave that I become sad'. Either way, I don't seem to get close
Sorry, just wondering, but how are you bilingual French and not know the meaning of 'laisser'? Because it has exactly the same meanings as 'lasciare': e.g. "laisse-moi expliquer" (FR) = "lasciami spiegare" (IT) = "let me explain" (EN). English even uses "laissez faire" (let [them] do [it]) as the name of an economic policy.