You're right that the plurality of "any" and "they" disagree. People often use "they" as a gender-neutral singular pronoun, but with "they'll all," it's clear that it's meant in the plural sense. "They'll all" refers implicitly to all doctors based on the context, but is not meant specifically to refer back to "any doctor," and consequently does not need the same plurality. Splitting the sentence into two sentences, or even just replacing the dash with a semicolon, would be a good way to make it even clearer that the antecedent of "they" is not "any doctor."
Ask any doctor; they'll all tell you that alcohol is a poison.
Ask any doctor. They'll all tell you that alcohol is a poison.
If the antecedent of "they" cannot be "any doctor" because of the plurality mismatch, what is it? It is not specified. Because of this, even though the meaning of "they'll all" is clear, the grammar is not perfect.