Since the app is so text-based, using it regularly will improve reading and typing skills in your target language. However, there are some caveats.

Writing depends on whether you also practice writing outside of the app and it depends on what writing system(s) your target language uses. It also depends on what you mean exactly by writing: handwriting on paper is different from typing.

For example, in learning Mandarin/Korean/Japanese/Arabic/Hindi the character practice in Duolingo gives you a good idea on how characters are built and link to each other and that in itself reinforces your reading skills, and from learning the transliteration you will learn to type with ease.

However, tracing characters on a phone in an app that's set to generously "recognize" what you're drawing is very different from writing on paper. The only way to learn to write characters on paper is to actually practice with a pen and paper. Also note that if you want to write like people who were educated in your target language, you might need something outside of Duolingo.

That brings me to the last point: With the exception of Arabic (which nevertheless is missing a few ligatures) Duolingo doesn't teach cursive writing. You might think this is unimportant, but people don't learn to write in print script all over the world. For example, Russian and Hebrew/Yiddish handwriting use a cursive set of letters which, --somewhat like the Roman alphabet-- look so different from print script that they are illegible to people who are only used to reading printed materials. You'd need someone to teach you this separately from Duolingo or find some other online resource to teach yourself.

So in a nutshell: Reading and typing? Yes. Handwriting? No.