Are you sure that you didn't misread the sentence - or that it was a typo? As a native speaker, I have never heard of bieren. I would consider it incorrect.
As for other drinks, it is also not possible. There are, however, some constructs that at least sound like a drink would be in them, even if they mean something different:
- ich weine = I am weeping
- ich wässere = I am watering (e.g. plants)
Update after reading the PDF:
I suppose, this is the PDF for "Grammatik und Design". The sentence used is in page 3:
Außerdem entstand auch in einigen Fällen der Neologismus „Ich biere
The text explicitly states that the sentence in the question is a neologism, i.e. it is a language construct that has been newly formed. Many neologisms are not considered correct for quite a while, yet they can be used to command the attention of the listener/reader. Maybe, after a while, this construct then will be considered correct. An example for this would be another advertisment-related neologism "Hier werden Sie geholfen", which is also not considered correct, but is used in noncommercial context and originated from an ironic manner.