We know by Faraday's law that emf induced in a closed coil is given by the negative rate of change of magnetic flux associated with the coil....but is the induced electric field set up in any coil due electromagnetic induction always non conservative....for example, the one setup due to self induction or due to mutual induction?
An electric field will be conservative or not depending on its curl
A field with zero curl can be made with an arrangement of electrodes. If you take a point charge, it will gain and lose energy as you move it around. However, if you take it on a closed loop, the energy will be the same before and after the transit round the loop. The potential energy of the point charge at some location is another way of defining the electrical potential there.
A field with non-zero curl can be made by having an area of changing magnetic flux through it. If the loop path taken by the point charge misses the area of flux, its movement will be conservative. If the loop path encircles the flux, then its movement will be non-conservative, it will gain or lose energy in the motion. If the point charge makes several rotations around the flux, it will change energy each time it passes round. The potential of the field in this case is multi-valued, depending on how many times we have been round the flux.