How heat flow through finite temperature drop is an irreversible process? Is it necessary that the drop/rise in temperature by flow of heat is irreversible?
 A: 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.
A: 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.
A: 
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.  
A: Suppose we have a hot body at temperature $T_H$ and a cold body at temperature $T_C$. Heat will flow from the hot body to the cold body. What would it take to reverse this process? We would need a refrigerator or a heat pump to move energy in the opposite direction from the cold body to the hot body. However, recall Clausius' statement

It is impossible to construct a device which operates on a cycle and produces no other effect than the transfer of heat from a cooler body to a hotter body

Therefore, this heat pump would require some work from the environment during the reverse process, and the surroundings will be modified during the process. But recall that the definition of a reversible process is one in which the system undergoes a process and is returned to its initial state with no change in the surroundings.
Therefore, heat transfer through a finite temperature difference cannot be reversible.
Now, suppose that $T_H=T_C+\Delta T$. What is the work required from the heat pump as $\Delta T \to dT \to 0$? The work must approach $W \to 0$, in which case the work required from the environment is zero, and the process is reversible.
From Fundamentals of Thermodynamics by Sonntag,

A heat-transfer process approaches a reversible process as the temperature
difference between the two bodies approaches zero. Therefore, we define a reversible heat transfer process as one in which the heat is transferred through an infinitesimal temperature difference. We realize, of course, that to transfer a finite amount of heat through an infinitesimal temperature difference would require an infinite amount of time or an infinite area.
Therefore, all actual heat transfers are through a finite temperature difference and hence are irreversible, and the greater the temperature difference, the greater the irreversibility. We will find, however, that the concept of reversible heat transfer is very useful in describing ideal processes.

