What is the significance of commutator relationships in physics, e.g. $qp-pq = i \hbar$, $R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z$, etc? Quantum mechanics has the commutator relationship: $$qp-pq = i \hbar$$
In relativity the Riemann tensor is a measure of how much covariant derivatives along a path commute. $$ R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z .$$
In thermodynamics it is essential to distinguish the exact and the inexact differentials apart; e.g. $đW,đQ $ vs $dS,dV, dT$ etc. which is based on the commutativity of the partial derivatives I believe.
$$ \frac{\partial^2 f}{\partial x \partial y} \neq \frac{\partial^2 f}{\partial y \partial x}.$$
Are these coincidental? Why are these so important? Are there analogies, or underlying concepts? Is it a reflection of the properties that these objects share?
Are there more commutator relationships in physics, e.g. classical mechanics?
 A: Differential forms
For the Riemann tensor and the thermodynamics quantities, the common point is that those object are differential forms on a manifold (for the Riemann tensor, this is not explicit in OP's notations, but you see curvature form or spin connection for more details). The mathematics of those differential forms relies heavily on antisymmetric/antisymmetrized quantities, which gives it this "commutator" flavor.
There are many more examples of differential forms : the electric field $E$ is a $1$-form, the magnetic field $B$ is a $2$-form and they are united in the Faraday tensor $F = E\wedge \text dt + B$ which is a $2$-form. More concretely, Maxwell's equations involve the curl (which is dual to the exterior differential on $1$-forms) and equations like :
$$-\partial_t B_z = \partial_x E_y  - \partial_y E_x$$

Why are differential forms  important in physics ? In physics, we know understand that the laws of nature should depend on the coordinate systems we use. This makes differential geometry and the theory of fiber bundles very useful, because they allow us to define and manipulate coordinate invariant objects.  Differential forms are such objects, and are very versatile, as they allow for both differentiation, integration and some algebra.

Lie and Poisson bracket
The commutator of linear operators on a Hilbert space $[q,p] = i\hbar$ from the canonical commutation relations in quantum mechanics is different. As @DiracDeltaYeah pointed out in the comments, it has a classical analog, namely the Poisson bracket :
$$\{f,g\} = \frac{\partial f}{\partial q}\frac{\partial g}{\partial p} - \frac{\partial f}{\partial p}\frac{\partial g}{\partial q}$$
which has $\{ q,p\} = 1$. (The Poisson bracket also has a relation to differential forms, in the context of symplectic geometry). Quantization involves the replacement $\{\cdot,\cdot\}\rightarrow \frac{1}{i\hbar}[\cdot,\cdot]$ of the Poisson bracket by the commutator of operators.
The commutators of quantum mechanics are also deeply linked with the Lie brackets from Lie algebras. Those represent the way infinitesimal transformations articulate with each other. For example, the commutation relations for spin/angular momentum operators :
$$[S_i,S_j] = i\hbar \epsilon_{ijk} S_k$$
is (a representation of) the Lie bracket of $\mathfrak{so}(3)$, the Lie algebra of infinitesimal rotations in $3$d space.
There are also Lie brackets of vector fields on a manifold, which are infinitesimal transformations in this context also. The formula is :
$$[X,Y]^\mu = X^\nu \partial_\nu Y^\mu -  Y^\nu \partial_\nu X^\mu$$
Those appear in general relativity (see for example Killing vectors)
In the context of non-abelian gauge theories, both differential forms and commutators appear : here the potential is a $1$-form $A$ with values in a Lie algebra. The field strength, covariant derivative and gauge transformation involve a combination of differential form calculus and Lie algebras.

Why are those forms of commutators and brackets important in physics ? I feel like the answer is clear here. The Poisson bracket plays a central role in Hamiltonian dynamics, commutators of operators is central in quantum mechanics and Lie brackets are central to the theory of symmetries.

