The normal force acting between two bodies during elastic collision is conservative or or non conservative?

In a theoretical universe it is conservative. If you have a perfectly elastic collision this is true, however that is fairly dificult to even approximate in real life.

First of all for an elastic collision you need the objects/materials that are colliding to only deform elasticly. Rubber bands do this quite well over a large range, however they dissipates part of the potential energy/work (from you stretching it) as heat (you can feel this if you stretch it quickly and often in a short time). Metal and glass can also deform elasticly but only in a much shorter range. Dropping a steel marble (ball-bearing) from a small height on a hard floor will allow it to bounce like a bouncing ball, same with a glass marble. They will actually bounce up higher if dropped from a small height in comparison to a rubber bouncing ball.

Why because rubber deforms much more and converts much more of the energy to heat and a bit of sound, some of the energy is also lost to vibrations in the ball. This will generally always happen in the real world as true elastic collisions don't exist as far as we know.

I also don't understand what the problem is exactly regarding the normal force. There's no innate relation between the normal force and an (true) elastic collison. A normal force is nothing more than a force generated on an object because it's resting on a surface/plane. Like if an object sits on the floor, it's the result of the floor pushing on the object due to gravity, if there was no floor it would keep on accelerating. For an object to not have any changes in it's velocity (vector) the net result of the forces should be 0, so for an object to not move or sit still it can not accelerate.

However when an object falls, it's better to address it as an impact force as that makes much more logical sense. if you have two blocks/balls moving a long a surface horizontally then the normal force is perpendicular to the collision and doesn't contribute. In reality friction would exchnage movement/speed into heat when an object moves over a surface. However when people are discussing these things such as true elastic collisions, they assume friction is non-existent. Nonetheless in this case friction, nor the normal force do anything to interfere with the collision.