A paradox to Lenz's law I have read that in simple words, Lenz's law states that:

The direction of current induced in a conductor is in such a fashion, that it opposes its cause.

This validates law of conservation of mass-energy.
I arranged the following thought experiment:
Let there be a pendulum with its bob being a small bar magnet. The pendulum is oscillating in a direction parallel to the horizontal axis of the bar magnet on which the North and South poles lie. Also, the pendulum is in complete vacuum. (But gravity is there to make the pendulum oscillate.)
At one of the extreme positions of the pendulum, we keep a solenoid, ends of which are connected to a load resistance.
As the North pole of the bar magnet approaches the solenoid, current is induced in the solenoid in such a fashion that a North pole is formed at the end near to the bar magnet's North pole, and the bar magnet gets repelled towards the other side.
The bar magnet then goes to the other end and then comes back (as a pendulum does) and again the same process is repeated. This should go on forever, and current should keep appearing across the load resistance.
How does the law of conservation of energy hold here?
 A: 
This should go on forever, and current should keep appearing across
  the load resistance.

This is a contradiction.  Since there is current through (not across) the load resistance, there is work being done on the load:  $p = i^2R$.
Let's be clear on this:  the coil-load system does no work on the pendulum, the pendulum does work on the coil-load system.
As the pendulum approaches the coil, the induced magnetic field repels the magnet; as the pendulum recedes from the coil, the induced magnetic field attracts the magnet.
Thus, with each cycle, energy is transferred from the pendulum to the load; mechanical power is converted to electrical power which is converted to heat by the resistor.
A: 
I've made a small illustration depicting the key idea. If this is in coherence with what you've asked, we could summarize some important points about the case.

Thought experiment http://imageshack.com/a/img69/807/u337.png
Total energy of the Earth and Bar Magnet system is given by the equation:
$KE + PE = \frac{1}{2}mv^{2} + \frac{GMm}{R}$
While PE is there for both Earth and magnet system (combined), KE is available for the magnet to spend on oscillation. If, while oscillating, it spends this KE by transforming it in to thermal energy in the resistor-inductor circuit, KE decreases slowly and becomes zero.
$KE + PE = 0 + \frac{GMm}{R}$
ie, the oscillation seize to exist.
An easy way to look at it would be to consider change in KE. We will start with the magnetic pendulum:
Component of angular acceleration = $\alpha =\frac{-Lmg\ sin(\theta)cos(\theta)}{I}$
This clearly has its peak values at 45 degree amplitudes. Now, component force due to this restoring torque is give by,
$F =\frac{-L^{2}m^{2}g\ sin(\theta)cos(\theta)}{I}$
From this pushing or pulling force we can derive the power and thereby energy:
$P=Fv$
$\frac{dE}{dt} = Fv = \frac{-L^{2}m^{2}gv}{I}sin(\theta)cos(\theta)$
On integrating,
$\int dE = \int \frac{-L^{2}m^{2}gv}{I}sin(\theta)cos(\theta)\ dt$
$\int_{0}^{\frac{T}{2}}dE = k\int_{0}^{\frac{T}{2}} vsin(\theta)cos(\theta)\ dt$
This gives the push energy in one single push (half the oscillation period).
Now the energy burned by the resistor is given by:
$dE = Pdt$
$\int_{0}^{\frac{T}{2}}dE = \int_{0}^{\frac{T}{2}}Pdt$
$\int_{0}^{\frac{T}{2}}dE = \int_{0}^{\frac{T}{2}}I^{2}Rdt$
Current in the LR circuit is
$I = \frac{V}{R}\left (1-e^{\frac{-Rt}{L}} \right )$
Substituting for current,
$\int_{0}^{\frac{T}{2}}dE = \int_{0}^{\frac{T}{2}} V\left (1-e^{\frac{-Rt}{L}} \right )^{2}dt
$
Thus the change in energy for a half cycle is
$\Delta E_{\frac{T}{2}} = k\int_{0}^{\frac{T}{2}} vsin(\theta)cos(\theta)\ dt - \int_{0}^{\frac{T}{2}} V\left (1-e^{\frac{-Rt}{L}} \right )^{2}dt$
This is how the RHS dampens energy change. The resistance in the second integral dominates in each integration. If it wasn't there, The remaining emf/potential term will cancel itself in a full cycle integration. Thus, there would not be an energy change. This isn't surprising because inductor can give back the magnetic force in each half cycle with the same magnitude. I hope this will make it more clear since my previous attempt was somewhat dubious.
A: The short answer is that there is a induced force on the magnet. This induced force will make the pendulum loose energy in the same proportions as there is electrical energy being generated.
A good experiment to show this effect is by having a small bar magnet and a copper pipe Or solenoid.
When you let a small bar magnet drop from a certain height it will hit the ground with maximum kinetic energy proportional to that height. However if you take a copper pipe or solenoid and you let the magnet free fall inside it there will be a current generated like you said in the OP. But there will be a force on the magnet slowing it down. which means you will notice it reaching the bottom of the pipe / solenoid  latter than it would have in complete free fall. 
A different way to look at it is with they law of momentum.
You could use the voltage across the load resistance to accelerate a different magnet. 
Which means you are giving that magnet momentum.
Which in order to be conserved should have been lost by the first pendulum. Show there was a  feedback force.
A: Actually pendulum wont oscillate forever.Its energy turns into heat in resistor.In other word it's domain would be something like this figure. 
A: As the magnet approaches the solenoid, a current is induced. The current generates a magnetic field. The field repels the magnet, slowing it's approach. The amplitude of the oscillations diminish. 
If there was no resistance, this would work in reverse as the magnet receded from the solenoid. The magnetic field would accelerate the magnet. The magnet would induce a current in the other direction, reducing the current to 0. This would reduce the field of the solenoid to 0. The amplitude would not diminish. 
A: The force on the pendulum only applies when the pendulum is in the vicinity of the coil. At that moment the harmonic motion of the pendulum is distorted. It 's amplitude is lessened and with it the upward motion. So the kinetic energy of the pendulum is converted into gravitational energy and electric energy. But the gravitational energy is less than without the coil present. The total mechanical energy (gravitational and kinetic)  is reduced. The pendulum swings back but next time it comes around the velocity in the equilibrium is less, the change of the Flux per unit time in the coil is less, so less energy is converted into electrical energy, but the gravitation energy is still reduced. This continues until the amplitude of the swing is that much reduced, that no flux is induced anymore in the coil. 
Furthermore when the magnet pulls away from the coil the flux in the coil recedes, the current in the coil turns around to form a attracting force, reducing the velocity of the pendulum even more, while the kinetic energy is converted to electric energy and from there into heat in the resistor. 
A: When the bar magnet approaches the solenoid, the magnetic field oppose the cause and repel it.
      But when it is departing, the induced magnetic field will oppose that too. Thus magnet will be attracted to solenoid which decelerate its motion. It will be exactly equal to the previous acceleration.
     Therefore the solenoid have no effect on its motion and it will die out due to frictional forces.
