From the book Quantum Mechanics by Cohen-Tannoudji it seems that the only requirement for an Operator to be an Observable is to form an orthonormal basis in the state space (finite or infinite dimensional)
"By definition, the Hermitian operator $A$ is an observable if this orthonormal system of vectors forms a basis in the state space."
In the rest of the book, the Hamiltonian $H$ is always referred to as an Operator (e.g. when talking about the Schrödinger picture). However I fail to see why in some cases the Hamiltonian cannot fulfil the definition of Observable.