Establishing a relation between different "entropies" is less trivial than it appears from the choice of the same naming. Unfortunately, even in scientific papers and textbooks, it is quit easy to find careless and unjustified identifications without specifying their limits.
Information theory provides a very general expression for the average information to be assigned to a probability distribution $\{p_i\}$, the Shannon's formula:
$$
H_s = -\sum_i p_i \mathrm{log}p_i.
$$
Due to its similarity with the Gibbs-Boltzmann statistical mechanics expression, this average information was also named Information Entropy. One of the reasons is that it reduces to the Gibbs-Boltzmann entropy (a part a trivial constant factor depending on the choice of units) if the probability distribution coincides with any of the statistical ensemble distributions of Statistical Mechanics. Therefore, at the thermodynamic limit, one can get contact with thermodynamic entropy.
A key step in order to establish link to equilibrium thermodynamics is then played by the probability distributions of the statistical ensembles describing system at thermodynamic equilibrium. It is a consequence of Liouville's theorem that any equilibrium probability density of Hamiltonian systems must depend on the microscopic dynamical variables trough the Hamiltonian and that is precisely where the special link between thermodynamic entropy and energy is established (do not forget that thermodynamic entropy differences are routinely measured with calorimeters!)
Shannon's formula is actually much more general, since it does not require a Hamiltonian system, and in principle not even a stationary probability distribution.
So, it could be applied to encode into an information entropy the information of a probability distribution of outcomes, provided a macroscopic state has been selected by a wise choice of an macroscopic observable like the total number of heads.
However, there is no natural way to make contact with the usual thermodynamics. First of all, there is no natural way to speak about equilibrium. The main reason being the lack of an underlying dynamics. Let's notice that the existence of such a dynamics would be a necessary but not sufficient condition for the existence of a thermodynamic-like entropy. A second, equally important, condition would be that the macroscopic observable(s) used to characterize the statistical ensemble should be preserved by this dynamics. The proposed example of coins bouncing around in a washing machine does introduce a dynamics, but does not preserve the observable "number of heads". Nevertheless, it could be suitably modified. For example, by adding an automatic control which counts a new microscopic configuration only when there is a given number of heads.
Notice that even in such a case, missing a physically meaningful energetic description of the macrostate, there is no way to make full contact with usual thermodynamics without additional hypotheses or further modification of the model.
