I = \sum_i -p_i\log p_i
is a function of probabilities $p_i$ and although it is often called entropy, it is not the thermodynamic entropy of Clausius (that $S$ from thermodynamics defined through $\int dQ/T$). This is not only because of absence of $k_B$, but also because in order to give $I$ value, one must put in probabilities $p_i$.
No probabilities occur in classical thermodynamics, hence it is not possible to derive the above formula from thermodynamic laws.
However, there is a connection between $I$ and thermodynamic entropy $S$. This connection is: if a system is in equilibrium with reservoir so that it has volume $V$ and average of energy is $U$, a statistical estimation of its thermodynamic entropy $S^*$ (a function of $U,V$) can be calculated as the maximum possible value of $k_BI$ for all possible values of $p_i$ under the imposed constraints (volume is fixed to $V$, average of energy is $U$).
This rule was not, as far as I know, falsified for macroscopic bodies for which it is meant to be used. Why it is valid is not immediately clear.
The information theory comes in when we ask: what is the meaning of $I$ for arbitrary values of $p_i$? The answer it gives is: it is a measure of amount of data that is needed to exactly specify the microstate of the system given those probabilities.
With this interpretation of $I$ the connection can be rephrased in this way:
if a system is in equilibrium with reservoir so that it has volume $V$ and average of energy is $U$, the measure of uncertainty $I$ about the exact microstate given the macroscopic constraints $U$,$V$ is the same function of $U,V$ as thermodynamic entropy divided by k_B.
This relation has been verified for rarified gas and other simple cases and it is simply assumed it holds universally for any macroscopic system in thermodynamic equilibrium.