In classical propositional logic and in first order logic, if two formulas are logically equivalent then they are substitutable. That is, if we can prove $A \leftrightarrow B$, then we can substitute $B$ for $A$, for arbitrary formulas $A,B$.
(This result is proven in a number of places: Stephen Cole Kleene, Mathematical logic: Replacement Theorem, page 122; Peter Andrews, An introduction to mathematical logic and type theory (1986): Substitutivity of Implication, page 89 and Extended Substitutivity of Implication and Equivalence, page 94;Joseph Shoenfield, Mathematical Logic, Equivalence Theorem, page 34.
Are there interesting logics in which this is not the case? That is, where we have in the logic under consideration that $A$ is logically equivalent to $B$ but also that they are not substitutable for one another.
I should have been clearer. I am not interested in hearing examples from Epistemic logics (since this is one of my areas of research). The same is true of indexicals.
I am more interested, for example, in substructural logics where the property doesn't hold, or many valued logics, for example.