Lower bounds for learning in the membership query and counterexample model
Yes, some lower bounds are known. For example, assuming $NP \neq coNP$, you cannot even properly learn read-thrice DNF formulas in polynomial time.  There is a whole paper developing such hardness results using something called the "representation problem".

To answer your linked-to question: Schapire, in his dissertation, in addition to proving that "weak learning" = "strong learning," also improved on Angluin's bound and gave an algorithm that uses $O(n)$ equivalence queries and $O(n^2+ n \log m)$ membership queries for learning DFA.

One easy way to get lower bounds is information-theory.  You can figure out how many distinct targets there are and how many bits a query gives you, etc.  These upper bounds come close but aren't there.  There are also issues one needs to think about regarding how the "counterexamples" arrive to the learner.  A well-chosen counterexample can give away quite a lot of information.

Update to the discussion above: Angluin and Dohrn address the question learning with random counterexamples in a recent paper.