Do $LC$ tank oscillators violate the conservation of linear momentum? General setup
Consider the following thought experiment: A capacitor $C$ and an inductor $L$ are mounted with fixed joints on a plate in outer space at say $1$ meter apart. They are connected in the $LC$ harmonic oscillator configuration. When powered the energy is exchanged between one to the other one. The sum of the energies on both remains constant over time.
One can eventually even drop the existence of the plate and just assume that the capacitor and the inductor are at a fixed $1$ meter distance apart.
Question
Using the mass energy equivalence formula $E = m\cdot c^2$ follows that the mass of $C$ varies with the energy of the capacitor as well as the mass of L varies with the energy of the inductor. Is therefore the center of gravity of the assembly moving in time (oscillating) without any external force being acted on it? Does this violate the law of the conservation of linear momentum?
 A: 
Is therefore the center of gravity of the assembly moving in time (oscillating) without any external force being acted on it? Does this violate the law of the conservation of linear momentum?

Not even remotely. Circuit theory is an approximation of Maxwell’s equations. It is intrinsically non-relativistic, so using circuit theory in the context of relativity is inherently problematic. On the one hand you are ignoring relativity while on the other you are not. You are essentially guaranteed to get nonsensical results.
To do this properly you would need to use Maxwell’s equations, which are fully relativistic. If you analyze the circuit using Maxwell’s equations then you will find momentum is conserved. This has been proven in general, not merely for your specific case, by Poynting’s theorem.
A: It is a  theorem in relativity that the centre of "mass" defined by
$$
{\mathcal  E}X^\mu_{\rm cofm} = \int T^{00}(x) x^\mu \sqrt{g}d^3x
$$
does not move for  isolated system with total momentum zero. Here $T^{00}$ is the energy density and ${\mathcal E}$ the total energy
$$
{\mathcal E}= \int T^{00}(x)  \sqrt{g}d^3x
$$
Despite this theorem there are many appararent paradoxes that come from combining non-relativistic mechanics with the fully relativistic Maxwell equations. They are  resolved when you realize the the field momentum density
$$
{\bf p}=\frac 1{c^2} {\bf E}\times {\bf H}
$$
having $1/c^2$ factor means that one has to keep $1/c^2$ relativistic corrections to the mechanics.  The general heading for these effects is hidden momentum.
A: When capacitor shoots energy to the right, then capacitor recoils to the left.
