How heat flow through finite temperature drop is an irreversible process?

A heat flow through a finite difference of temperature is irreversible because it does not satisfy the definition of reversibility.

A thermodynamic process is reversible if an infinitesimal change of the external conditions reverses the process. To illustrate, let us consider a system at temperature $T$ in thermal equilibrium - that is at the same temperature - with a thermal reservoir. An infinitesimal increase of temperature $dT$ of the reservoir causes a heat flow to the body, which heats up by the same $dT$. If the external condition is now reversed, i.e. there is an infinitesimal decrease of the reservoir's temperature, then the heat flow also reverses, it goes from the body to the reservoir.

This will not happen for a finite temperature difference. Let us say the reservoir is $1+dT$ degree above the body. The heat flows to the body. Decrease the reservoir's temperature by $dT$ and the heat flow does not reverse. This is what meant by irreversible heat exchange.

Note that this definition is of crucial importance when dealing heat engines. This reversibility is what allows, for example, a Carnot heat engine also works as a Carnot refrigerator.

Think. Can heat flow from the inside of a refrigerator to a hot stove if you place an iron rod in that manner?

According to the thermal current equation, $$I = \frac{\Delta T}{R}$$ is always positive. For $I$ to be positive, the difference in temperature should be positive. That is, thermal current flows from a source of higher temperature to that of lower temperature.

The drop/rise in temperature by flow of heat. Is it necessary that it is irreversible?

It is always irreversible for every real physical thermodynamic process (this is any thermodynamic process that is not a 'theoretical ideal' one, which don't exist in reality). So your question doesn't really make sense.

However, since the entropy of every real physical thermodynamic process is always greater than zero, this could be a thought upon as a 'necessity' that heat flow is irreversible.