Some of the operations that are performed on the tensors in a neural network are non-linear. An example might be taking the tanh of all of the elements of the tensor. For these steps, you won't have invariance (or covariance) under change of basis.

Even in physics, there are applications of tensors which essentially treat tensors as multidimensional arrays (see for example, tensor networks) with no predefined transformation properties. But the operations done on tensors are always linear.