Translation:You will not tell her; otherwise, she will tell the boss.
tu diras is the simple future tense ("you say/tell", "you will tell/say"). It is negated here and Duo has translated it "you will not tell (it to her)".
It is hard to imagine an English translation that would use the future tense in that form, it is clearly an order, which in English would be "Don't tell (it to) her". I can't see any other sensible way to translate this.
"Lest" is indeed uncommon but it is the right word for this sentence. I did not understand the sentence's meaning until I read BrunoGiuff's translation: "You will not tell her; otherwise, she will tell the boss" = "You will not tell her; if you tell her, she will tell the boss" = "You will not tell her, lest she tell the boss".