When two objects with a potential difference existing between them are connected by a superconductor, will the potential difference cease to exist?

The question is a bit vague, so here's a couple answers.

You are correct that all free carriers, be they electrons or holes, would travel to balance out the potential.

However, plenty of materials such as dielectrics, hold-off voltages from external interfaces, so the electrons can't easily travel to the point of contact with the superconducting connection. That's how a capacitor holds energy even tho' there's a potential difference between the electrodes.

So even tho' the superconductor has no resistance, I think the dielectric itself has internal resistivity. This means some energy might be dissipated there (and the discharge would take longer time than if you simply connected two charged metal plates).

In my answer I am referring to perfectly ideal case -- no resistance of any kind at all.

First of all,no voltage is needed to maintain a zero resistance current.Superconducting materials have been known to maintain currents for years by simply connecting two ends of the wire. This means that energy of cell is not getting dissipated out of the system as heat.

When the current passes through a resistor,energy of cell is dissipated out of system as heat,thereby reducing continuously.This is the reason why batteries get exhausted after sufficient use.

What I am trying to say the current will flow with or without any battery. Voltage plays no appreciable role here.It will not cease to exist.Rather it will continue to exist without any difficulty.It would be comparable to the stage when the battery was kept in the corner without any connections to it.

Hope this helps.

We know that voltage/potential difference is the amount of work done or energy released by the battery/cell for transferring a unit charge for higher potential to lower potential.

To be more precise, the potential difference (voltage) between two points is the work required per unit charge to move the charge between the two points.

Now, when we connect two point with potential difference by a superconductive wire, there remains no resistance.

If there is no resistance between the two points (superconductive wire) then no electrical work is required to move the charge at constant velocity (constant current) between the points. There is no potential difference.

A mechanical analog is an object sliding at constant velocity between two points on a perfectly frictionless surface. No mechanical work is required to move the object between the two points.

So when electrons traverse through that wire, heat/energy released or work done should be zero. So work done or energy released per unit charge i.e. voltage also becomes zero. So, will the potential difference cease to exist?

It's not that there is a potential difference in the superconductor that ceases to exist. It never existed. In both the above examples, if the charge (or object in the mechanical analog) was originally at rest, some work had to be done for it to acquire the kinetic energy associated with its velocity from somewhere, per the work-energy theorem. However, once the current (velocity) is acquired, no work is needed to keep the charge in motion.

Hypothetically, let's say a battery is connected across the superconductor before any current flows. All real batteries have internal resistance. The current flow in the superconductor will equal the battery internal emf divided by its internal resistance. The potential difference between the battery terminals is zero since all the emf drops across the internal resistance. You now have current flowing in the superconductor without a potential difference across it. Theoretically if you could somehow simultaneously replace the battery with another supper conductor to maintain a complete circuit, the current would continue to flow.

Hope this helps.