The inertia tensor seems like it cannot depend in any way on position, but every other tensor in physics is a tensor field (stress tensor, electromagnetic tensor...) so, which is it?
Tensors do not have to be tensor fields. Similarly, scalars do not have to be scalar fields, and vectors do not have to be vector fields.
The inertia tensor is defined very specifically and is related to the distribution of matter about a set of axes in space (which can be chosen to move with the object's CoM). This definition produces a matrix (or tensor) by integrating over the entire body, the entire mass distribution. As such, once the evaluation is complete it cannot possibly depend on position within the matter distribution (or outside it). The situation is different for the strain tensor for example. Since small elements of matter can be compressed by different amounts that will be a field in general. However, at any instant the MoI will be a matrix. The same could be said for the CoM of a soft object. As it moves the mass elements will continuously redistribute in space and this may cause the CoM and MoI to require being calculated at each instant but the CoM will always just be a single vector (not a vector field).
Now, based on the above, it is pretty clear that these quantities can depend on time.
The inertia tensor is just a tensor, not a tensor field.
It is also not the only example in physics of a tensor that isn’t a tensor field. A slightly more exotic example of such a tensor is the polarizability tensor, that describes the tendency of a material to become electrically polarized in the presence of an electric field. For materials that are anisotropic, the direction of the polarizability vector isn’t necessarily the same as that of the electric field vector, because the material has a different tendency to become polarized along different axes. The polarizability vector encodes the relevant information about how the material will behave. A good discussion of this can be found in the Feynman lectures.
In general, tensors over a three-dimensional space are useful for describing the properties of anisotropic materials. For example, there is also the Cauchy stress tensor in continuum mechanics. But that’s a tensor field.