Q:

Tensors and rotations

All the tensors that I have studied so far have always appeared with some kind of rotation.  For example, spherical tensors rotate as spherical harmonics, tensors in the context of special relativity transform via the Lorentz matrices that are just rotations in the the 4 dimensional space-time.  My question is the next, do all objects that are called tensors always have to have some kind of rotation associated with it? 

A:

There is a rather general notion of tensor in physics for which the answer to your question is yes provided you replace the word "rotation" with "group element."   In particular, the notion of a tensor that is often used in physics is not restricted to that of multilinear maps on manifolds (which naturally have a particular transformation law under coordinate transormations). 
Notice that both the set of rotations, and the set of Lorentz transformations form groups under matrix multiplication.   The group of rotations on $3$-dimensional Euclidean space, $\mathbb R^3$, is called $\mathrm{SO}(3)$.  The groups of Lorentz transformations on Minkowski space $\mathbb R^{3,1}$ is called $\mathrm{SO}(3,1)$.  So when we say that a tensor is an object that transforms in a specific way under rotations, we are specifying that the object transforms in a specific way when acted on by elements of certain groups.   For example, a $(k,0)$ tensors under rotations transforms as
$$
  T^{i_i\cdots i_k} \to R^{i_1}_{\phantom{i_1}j_1}R^{i_k}_{\phantom{i_k}j_k}T^{j_1\cdots j_k}
$$
We can generalize this to arbitrary groups as follows.   Let $G$ be a group, and let $\rho$ be a representation of the group acting on a vector space.   This means that $\rho$ assigns a linear transformation $\rho(g)$ on the vector space to each group element $g$.  In a given basis for the vector space, we can write the matrix representation $\rho(g)^i_{\phantom ij}$.  Then we can define a $(k,0)$ tensor with respect to the representation $\rho$ as an object $T^{i_i\cdots i_k}$ that transforms as
$$
   T^{i_i\cdots i_k} \to \rho(g)^{i_1}_{\phantom{i_1}j_1}\rho(g)^{i_k}_{\phantom{i_k}j_k}T^{j_1\cdots j_k}
$$
Notice that tensors under rotations and Lorentz tensors are a special case of this definition when the representations in question are those that map each rotation to itself and each Lorentz transformation to itself (often called the defining representation).