- Is classical electromagnetism a dead research field?
- Are there any phenomena within classical electromagnetism that we have no explanation for?
J. D. Jackson in the introductory remarks of his chapter on 'Radiation Damping, Classical Models of Charged Particles' (3rd edition), says that the problem of radiation reaction on motion of charged particles is not yet solved. He says that we know how to find motion of charged particles in given configuration of EM fields and also how to calculate EM fields due to given charge and current densities. However, when a charged particle accelerates in a field, it also radiates and we usually ignore the radiation reaction.
You can also refer to the paper
If you consider plasma physics and magnetohydrodynamics to be part of classical electrodynamics your list of open problems may grow.
There are still some very important open problems in the classical electromagnetism of relativistic charges, and there is indeed no satisfactory resolution of the reaction force and self-field problems for a relativistic point charge. One good resource for this is
They identify five main open questions in classical electromagnetism. From their preface:
As orbifold mentions, it is tempting to dismiss most of these as problems that are naturally solved in the transition to quantum electrodynamics. However, that neglects the fact that QED involves renormalization procedures which are also pretty hard to swallow. I tend to think that exploring the classical side of the problem is one good place to look for ideas into how the quantum side might be improved.
Applied electrodynamics isn't dead, especially when you include light: optical tweezers, orbital angular momentum multiplexing, electromagnetic simulation of high-speed PCBs etc.
One of the main remaining problem in theoretical classical electrodynamics was an adequate expression for the self force on a radiating charge without pre-acceleration and runaway solutions. Over the last decade in particular, this has been largely solved, as claimed by Rohrlich$^1$:
 Fritz Rohrlich, Dynamics of a charged particle, Phys. Rev. E 77, 046609 (2008)http://arxiv.org/abs/0804.4614