Translation:I like beer, but I do not drink it.
The same happens in Portuguese: saying 'eu não bebo' implies that you don't drink any alcoholic beverage. This specific phrase isn't clear at all, but this is a situation in which it's hard to make yourself understood - '... mas eu não a bebo' (literally, 'I don't drink it') would sound too much formal. The only exit I can see for this problem would be using some more words for denote intensity, as 'Eu gosto de cerveja, mas eu não bebo muito' ('I like beer, but I don't drink too much') or 'Eu gosto de cerveja, mas eu não costumo beber' ('I like beer, but I'm not used to drinking').
That's right, both "eu" are implicit and thus not really necessary in this case. @pezbabel's answer is actually a bit more common than the default answer.
The use of the pronouns in these cases is a matter of "feel" and depends on the context. For example, if I am enumerating all the things that I like, it is clear that all the sentences refer to "eu", so the pronoun is not necessary. But if I'm in a roundtable and everybody's saying what they like and someone turns to me and asks "what about you?" then I would likely start the sentence with "eu".
"I like the beer, but, I don't drink" sounds like there is a beer can in front of you and you're referring to that specific beer; besides, you never put a comma after the "but". As you're saying that you like the beverage beer in general, you don't use the article. This applies to anything you say you like, dislike, hate, want to consume etc.
Obs.: "I like the beer" would be "Eu gosto da cerveja".