Consider the simple case of a battery in series with a resistor, mediated by conducting wire.
At steady state, the charge carriers are circulating around the circuit at constant speed through the conducting wires. From this we can conclude that the battery is doing no work on the charge carriers (or else the charge carriers would gain kinetic energy), and the entirety of the battery's energy output is being dissipated by the resistor in the form of heat ($V I$ watts).
Edit: It has been pointed out by multiple responses that this point is incorrect. It is more proper to say that work is done on the charge, and then negative work is done on it again inside the resistor. However, I think we can still argue that electric field is 0 inside the conducting wire and continue with the paradox.
Let $E$ be the electric field and $\gamma$ be a path completely contained in the circuit and that begins and ends inside a conducting wire. Since the battery does no work on the charge carriers, it follows that $\text{work} = \int_\gamma E \cdot dl = 0$. In particular, $E \equiv 0$ at all points in the conducting wires.
Edit: To fix the argument here, I should say that the electric field is 0 inside the conducting wire simply because charges do not accelerate at any point within the conducting wire.
Since the circuit has a net neutral charge overall, there is no electric field in the ambient space outside of the circuit.
The only nonzero electric field is contained within the battery and resistor themselves, and their fluctuations are responsible for the net increase in kinetic energy of the particles inside the resistor.
Now add in an uncharged capacitor in parallel with the resistor.
Initially, the total electric field between the plates of the capacitor is the one produced by the circuit in the ambient space, which was found to be zero. And since there are no forces which cause charge carriers to accumulate on one of the plates, the field between the plates will remain zero.
This state of affairs contradicts circuit theory, which predicts an electric field will form between the plates due to an accumulation of charge carriers on one of the plates.
What is the resolution?