Ya me cansé de que me tomes la cerveza has two instances of me. The first is trivial: it's part of the pronominal verb cansarse, which means "to tire, to become tired", so let's leave that alone.
The second me can be analyzed as a dative of harm construction (dativus incommodi). This is a form that has a parallel in the dative of benefit you can find in phrases like
Me tomé toda la cerveza.
Nos comimos una pizza.
I once answered to a question about reír(se) and other pronominal verbs that deals with this mediopassive voice form, and that you might find useful.
Let's go back to the lyrics of the song, but suppose (for brevity) that the sentence is just
Me tomaste la cerveza.
That meaning could be translated literally as "You drank *me the beer", or, rephrased correctly, "You took the beer from me and drank it", or simply "You drank my beer", only the former translation is too long and the latter misses the connotation of harm or loss; that's what the pronoun me is about. It's as if you could say "You drank the beer from me" in English.
The same meaning can be expressed (from the other side, as it were) as
Te tomaste mi cerveza.
In this case the translation is "You drank my beer", but the inclusion of te turns the phrase into a dative of benefaction: not only did you drink my beer, but you drank it for your benefit. The two sentences therefore form an opposing pair.
The benefactive construction is standard in Spanish. The dative of harm is common in Argentina and mostly informal.
There's a phrase that worried mothers often say to the pediatrician when their children refuse to eat properly: El nene no me come. If you run this past a translator it will come out "The child does not eat me." The meaning, of course, is "The child does not eat and that affects me."