Do I need to let ravioli made with fresh dough dry?
Counter-question: Do you think it’s a safe practice to let a filling that may contain raw meat or dairy hang around at room temperature or possibly even warmer for hours?
So no, you shouldn’t let homemade ravioli dry. Attempting to, even if you used a filling that is safe at room temperature, takes way longer than plain (and thin!) regular pasta. Commercial ravioli products are either quick-dried and sealed to make them shelf stable, or sold in the refrigerated section of your supermarket.
If you can’t cook and serve your ravioli immediately, either refrigerate them for short-term storage or freeze them.
And there’s no need to let plain pasta dry before cooking, one of the special charms of making fresh pasta is that it’s fresh, i.e. not dried. Drying is done for long-term storage. It allows working with larger batches (messing up the kitchen only once...) and having the pasta at hand without kneading, resting and rolling.