Why Lenz's law works? When we move a magnet towards a coil, current is induced in it. Magnetic field too develops around it. Since there is interaction of both magnetic fields(that of magnet and coil), there is magnetic potential energy in field(gained from the work we do). This energy is going to do work on magnet (either in repulsion or attraction). In this case Lenz's law tells us that it will be repulsive force.
If it is attractive or repulsive , the stored energy is to be spent on magnet and it does work on magnet. Also whole of energy we supplied is not used in this work as there is heat loss too. Then why we say it (in case of attractive force by coil on magnet) breaks conservation of energy?
 A: See if the other case was possible then just a little push (towards the coil) would give it a greater amount of kinetic energy .
Suppose we applied a force of 2N in moving the magnet towards the coil by a few centimetres ( let's say) 10cm then our work or energy transferred equals 0.2 J and this will give the magnet some kinetic energy and some part of this much energy is initially  transferred to the electrons from which little energy is  released due to the resistance of the coil . So what this means that the kinetic energy of the magnet should be less than earlier because it is moving towards the coil and it's energy is being transformed into heat and magnetic energy and it should stop after some time.
But if the magnet was attracted then it would have got some more kinetic energy . And the cause of that kinetic energy must be some work .
Since we left the magnet under attraction this means that the cause of that extra work was not us but the coil itself. But for inducing current in the coil very little amount of energy was supplied to the electrons , so how could this little energy amplify to such an extent without any external supply . Hence energy is created by the coil or say by induced current.
This means that energy is not conserved . Hence this scenario breaks energy conservation . And so it can't be true .
A: if the force is attractive, we only have to start the motion, then the attraction would accelerate it, the current would increase, and we get energy for nothing or a perpetuum mobile.
