"I don't drink beer."
Translation:Я не пью пиво.
'Hear' is not really the best word since «пива» and «пиво» are pronounced in the same way...
Why didn't you report it then?
Yeah, it is correct, just old-fashioned. Objects after negated verbs mostly took Genitive in the 19th century. Nowadays the choice of case is immensely more complicated.
For example, if you have an allergy and avoid drinking orange juice in spring, you'll say it «Я не пью апельсиновый сок» (апельсинового сока would immediately kick you back into pre-WWII era). Иметь (formal "to have") is one of the verbs that only use Genitive when negated. Abstract objects also tend to do this (e.g., не обращайте внимания ~ pay no attention). Style and the object being more vague might matter for some verbs. Complicated, as I said.
Пиво and пива are pronounced the same, however, so it will only look pretentious in writing.
Careful there! Sentences do not belong to cases. To give you an analogy, "This is my mom's best piece" has "my" and the possessive "mom's" but it does not mean that it is a "possessive sentence".
The Genitive is definitely used to negate existence (У меня нет кошки / У тебя не было кошки). However, for objects of transitive verbs, negation does not automatically trigger the Genitive (in modern Russian).
«Я пиво не пью» is used when 'beer' was mentioned before in the context. For example, when you were offered some beer and you're rejecting it.
«Я не пью пиво» is a 'default' word order when there's no extra context (it works well either when you're rejecting some beer, or when you're just describing what you like and don't like to drink). Duolingo doesn't have any previous context, so it sounds better here, at least to my ear.
First, words are not spelt phonetically in Russian (at least, not exactly). The spelling should be memorized. Probably, going from the spelling at the start makes even more sense for a non-native, since you will learn most words from books anyway.
Second, if you encounter such words in speech, you will one day know their grammatical gender or at least declension class. Neuter nouns end in о or е (ё). There are also ten neuter nouns that end in -мя (имя, время, пламя and all the way to вымя and темя).
Feminine -а-ending nouns end in -у in the Accusative, which does not seem to happen here.