It depends on the meaning of physical theories. E.g., classical electrodynamics and classical Newtonian are examples of physical theories that are not completely background independent. Of course, both these examples believed to be effective theories with limited range of validity. For example Classical Electrodynamics is not a valid approximation for the photo-electric effect, or quantum entanglement in optics.
So maybe the question should be whether a fundamental physical theory (Such that effective theories could imprinciple be derived from it) needs to be background independent?
First of all there is no logical necessity for a fundamental physical theory to be background independent. We can easily imagine fundamental physical laws for a hypothetical world with hypothetical inhabitants, which are not background invariant.
So the question remains, whether a fundamental theory describing the physical world we live in must be background independent. Ultimately, this depends on what the fundamental physical laws of our world are (which are yet unkown).
Our knowledge of the fundamental laws of physics is of course tied to empirical experience. So we can ask whether the yet known (non fundamental) theories indicate that true laws of nature are background independent. On one hand we have General Relativity which is empirically succesfull and shares a version of background indepedence. On the other hand, the empirically successful quantum theories do not share this feature. Some physicist believe that the feature of background independence in general relativity is essential, in the way that a successful theory succeeding general relativity most likely still has to feature background independence.
If a fundamental theory does depend on a background, the background should not affect in dynamics in a way contradict empirically verified predictions of General Relativity. But in principle, it is possible that deviations of background independent theories are beyond empirical access (which of course is a fuzzy boundary).
Another question is, what the precise meaning of background independence is. In the context of general relativity it means that the structure which is fixed a priory is just a manifold $M$ with its smooth structure, whereas other geometric structures like a Lorentzian metric $g \in \Gamma(T*M \otimes T*M)$ are determinded by the initial conditions and the laws of the theory. On the converse special relativistic theories usually are formulated starting with fixed Lorentzan manifold $(M,g)$, e.g. Minkowski space
Note however that there is of course still the background structure of $M$. Smooth manifolds encode a lot of mathematical structure, including the topology and the notion of smoothness (which is not in general determinded by topological manifold). So the meaning of background independence, depends on what kind of structure you count to the background. For example, does a theory with a fixed topology of the bare manifold $M$ count as background independent or not.
