First, a minor point: it doesn't make any sense to say that "a Hamiltonian commutes". You mean that the different terms in the Hamilonian commute. When it comes to commutation, it takes two (or more) to tango.
It is indeed often said that if all the terms of a Hamiltonian commute, then the Hamiltonian is classical. But that isn't really true in my opinion. It's true that if all the terms in the Hamiltonian commute, then there exists a basis of the Hilbert space that simultaneous diagonalizes every term. After you've found this basis and rotated into it, the Hamiltonian does indeed become effectively classical, because each term in the Hamiltonian just contributes a value (that term's eigenvalue) to the total energy, without changing the state at all. Since the Hamiltonian doesn't modify these basis states, its "operator" nature effectively falls away and it basically just becomes a scalar energy function.
But the energy eigenstates are typically complicated superpositions of computational basis states, so they can be nontrivial to calculate (and certainly to engineer experimentally). So the "quantum-ness" only vanishes after you've finished the highly quantum step of finding the simultaneous eigenstates of each term in the Hamiltonian.
For example, the toric code Hamiltonian has every term commuting, and you do sometimes hear the toric code described as "classical", but I personally would say that it's highly quantum.
Finally, saying that "all terms in the Hamiltonian commute" is somewhat ambiguous anyway, because you can sometimes group smaller non-commuting terms together into larger terms that do commute. There's no unambiguous definition of what counts as "a single term" in the Hamiltonian.
