Why electric potential requires absence of acceleration? As of 28/05/2019, Electric Potential is defined in Wikipedia in the article with the same name as:

The amount of work needed to move a unit of positive charge from a
  reference point to a specific point inside the field without producing
  an acceleration

What is the meaning of "without producing an acceleration"?
 A: The total energy of a body is the sum of kinetic energy and potential energy.
Kinetic energy depends on speed and potential energy depends on position.
The wiki text is a special case of the fact that if we move a body from A to B without altering the kinetic energy then its energy difference is only potential.
A: An accelerating charge radiates EM waves that carry energy,
the static electric potential difference does not account for the radiative exchange. 

To measure therefore the work done by the static potential difference while moving a charge from one place to another, must be performed so that the charge does not radiate.
A: I'm going to try a different approach to answer your question than my original answer.
The amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration
What is the meaning of "without producing an acceleration"?
This is not a good definition of electric potential, which is supposed to be synonymous with voltage. A better definition is "The potential difference $V$ between two points is the work per unit charge required to move the charge between the points" (re: NCEE FE exam reference manual for Electrical and Computer Engineering).
Secondly, the statement "without producing an acceleration" is misleading. Electrons are accelerated by an external electric field, but at the same time they are decelerated due to constant collisions with atoms and molecules in the material in which they flow losing kinetic energy in the form of heat. After each collision they once again accelerate. The end result is a constant average or "drift" velocity. 
Since the potential difference between two points is intended to only refer to a difference in potential energy between the points, I'm guessing the statement "without producing an acceleration" may allude to the fact acceleration would result in a change in kinetic energy of the charge between the points,  which is not part of the change in potential energy. But it's only a guess.
Hope this helps.
A: Let us imagine that we want to find the electric potential at a point $P$. To do so, we first select a reference point $R$. We then find the work done by us in moving a unit charge from $R$ to $P$. Let us understand the calculation of work in greater details. We apply a force $\vec{F}$ on the charge such that it just compensates the electric force on it. (The electric force comes from the electric field in the region. If there was no field then one would not find the potential.) Thus, $F$ has to be $-q\vec{E}$ and the work done will be
\begin{equation}
W = -\int_R^P q\vec{E}\cdot d\vec{x}.
\end{equation}
Since $\vec{F}$ just balances $q\vec{E}$ there is no acceleration of the charge. We say that the charge is moved `quasi-statically' from $R$ to $P$. Why do we need $\vec{F}$ to just balance $q\vec{E}$? It is because we want to apply just enough force to overcome the electric field and reach $P$ from $R$.
A: The work energy theorem states that the net work done on an object equals the change in its kinetic energy.
Therefore, in order for the work done on an object to only change the potential energy of object, the net work done on the object must equal zero;
"equal zero" means that no change in kinetic energy and therefore no acceleration causing that change.
Charge
In the case of charge, when work is done to move a positive charge between two points in opposition to the force the electric field exerts on the charge, that work only will result in an increase in potential energy if no net work is done on the charge which would cause an increase in kinetic energy.
The process is as follows:
The force exerted by an external agent moving a positive charge is in the same direction as the movement of the charge. Therefore that work done by the agent is considered positive and transfers energy to the charge.
Simultaneously the force on the electric charge due to the field is opposite to the movement of the charge. The work done by the field is therefore negative. In order for the charge to only acquire electrical potential energy the negative work done by the field has to equal the positive work done by the external agent in order that the net work done on the charge is zero. If it were less, the charge would have both potential and kinetic energy. 
Bottom line: For there to only be a change in electrical potential energy the field has to take all the energy given the charge by the external agent so that there is no net work done on the charge, and store it as electrical potential energy of the charge electric field system. 
The analogy with gravity is external positive work (say done by you) is needed to raise a mass initially at rest on the ground to a point at rest a height $h$ above the ground. Simultaneously gravity (whose force is opposite the motion of the mass) does an equal amount of negative work taking the energy you supplied the mass and storing it as gravitational potential energy in the mass earth system. Since there is no change in kinetic energy of the mass (and thus no net acceleration to cause such change), no net work is done on the mass just like no net work is done on the charge.
Hope this helps.
A: The definition of potential energy and potential(may be electric or gravitational) has lots of misconceptions and arguments related to it.I think that the introductory textbooks are faulty in providing the exact definition of potential energy.
Please consider these 3 points for potential energy:


*

*Absolute potential energy is not defined.

*Change in potential energy is well defined.

*Potential energy is defined only for a system consisting of 2 or more than 2  particles.


Please consider the definition below
The change in potential energy of the system is defined as the negative of work done by the internal conservative forces of the system.
This definition is the standard definition of potential energy of a system. This definition never says that acceleration is not allowed.
This definition gives us the whole authority to allow acceleration.
Some books provides its definition by use of external conservative forces acting on the system.
But these definitions have a fault in them because the work done by external forces on the system would be equal to change in potential energy of the system if and only if the change in kinetic energy of the system is zero.So these definitions consist of a specific case that acceleration must be zero.
So I think that you must have understood the definition of potential energy properly. Similar is the analogy of electric potential.
Consider these points for electric potential:


*

*It is defined for a system consisting of 2 or more 
than 2 particles.

*The charges due to which electric potential is 
calculated must be fixed.

*The test charge must be of small magnitude.


Electric potential for a system of charges is defined as the negative of work done by internal electrical forces of the system on the test charge per unit test charge.
Again,this definition allows acceleration.
Hope this helps!
A: If you rather let it go, the body will gain kinetic energy in expense of potential energy (due to conservation of energy).
When you move the body at constant velocity (i.e. no acceleration) you have to do work against the force producing the potential. This work done would be equal to the gain in kinetic energy it would have, if you had let it go. Hence, for energy conservation to hold, this must be equal to the loss is potential energy.
