...said no Czech person ever.
I said it :-)
Well, he/she probably means he's not drinking it at the moment. It's still more credible than 'Pivo já nepiju'. Right? ;)
In Czech there's no distinction between present simple and present continuous. So it can be both "I am not drinking beer" and "I don't drink beer".
"Já ti neverim," said the Irishman to the Czech. Sláinte!