Full load, voltage may less or more(+ve or -ve regulation) , it cannot be less all the time, it depends on load, with lagging(demagnetizing effect)  or leading(enhance field effect) power factor. Normally power factor of system is lagging and hence, terminal voltage, is less, at load....Read any book, on Electrical machines(alternator and transformer), with voltage and current vector diagram, with different power factor loading....

Because of winding voltage drop as current starts flowing....

If you have capacitive network you can have even higher voltage at the end of the line then full voltage (nominal or rated voltage) at the begining of the line-network.

If there is no load condition,  no current flows and there is no voltage drop in the series resistance, hence voltage across load is more than full load condition.

Ideally, if the voltage regulation is perfect, the voltage across the load should not depend on the load because of the negative feedback.

The ideal voltage source has an output impedance of 0 (zero) Ohms.

As there is nothing like an 'ideal voltage source' (this is just a theoretical model), you always have some voltage drop - caused by the load current on the non-zero output impedance. AND there is the impedance of the wires that adds to the output impedance...

One way to handle this non-ideality is to have separate voltage sense lines connected to the load. In this case, the voltage drop on the output impedance can be compensated.

Typically, voltage regulation is done by using some sort of servo loop. If the voltage at the load is too low, the output is boosted to compensate. "Too low" is determined by comparing the load voltage to a reference to produce an "error signal" which is used to determine the output.

For such a simple loop, increasing the output means there needs to be a finite error signal. That can be made small by increasing the loop gain but it can never be zero hence the full load voltage must always be less than off-load or partial load.

A more complex servo might use an integrator in the loop in which case the full load should be the same as off-load because the output of an integrator is constant for zero input, i.e. zero error signal. However, the limiting factors are often speed of response and stability so your servo design needs to reflect the requirements in any particular project.

An exotic way to achieve a zero error is to connect a negative resistor in parallel to the load:

https://www.maximintegrated.com/en/app-notes/index.mvp/id/815

http://www.circuit-fantasia.com/circuit_stories/inventing_circuits/par_nr_comp/par_neg_res_comp.htm

https://en.wikibooks.org/wiki/Circuit_Idea/Revealing_the_Mystery_of_Negative_Impedance#Compensating_resistive_losses_by_N-shaped_negative_resistors

Using that approach, the problem of stability remains though. If the gain is fractionally too high, the system will oscillate.

For DC circuitry this is quite natural situation  Some portion of voltage drops down due to internal resistance of battery (voltage source). Voltage decrease is proportional to E*(r/R), where E-electromotive force, r- internal resistance of battery and R-payload resistance. In case of AC current situation may be considerably complex and dependent on capacity/inductance of load and frequency of current. 

The terminal voltage when full load current is drawn is called full load voltage (VFL). The no load voltage is the terminal voltage when zero current is drawn from the supply, that is, the open circuit terminal voltage. Some portion of voltage drops down due to internal resistance of voltage source. Voltage decrease is proportional to E*(r/R), r- internal resistance of voltage source

That's true for an unregulated system, the question posed starts "In voltage regulation ..".

The regulation ceases to operate when load is disconnected, for most types of regulator, and a good thing it is, since the circuit is no longer a closed loop and regulation is normally not designed to draw negative current (see attached image of buck converter, for example). Having a little leaking current is frowned upon nowadays due to energy saving requirements.

If there is no signal error there will be no regulation, and the theoretical zero error in a PID regulator occurs after infinite time, at least not before load changes.

Stability is usually required, so there is a choice between small error or high bandwidth.

Filters (low pass) are added to eliminate noise (ripple and hum). Also in a switching converter. Even a tiny capacitor holds a charge and the accompanying resistance adds to the impedance.

All of these combine to decrease the output voltage under load.

There are essentially three types of power sources. 1. Constant Voltage 2. Constant Current and 3. Constant power. 

A Constant Voltage source, which is by far the most popular, is characterized by its ability to maintain the terminal voltage constant regardless of the current drawn by the load connected to its terminals. Batteries, Bench-top electronic regulated Power supplies, power supplies inside your computer, electric power generators etc are few examples of this. But all earthly systems have limitations and they are not ideal. That is you cannot draw infinite current for a practical, earthly constant voltage power supply. Even regulated power supply has Regulation specification. That is its output voltage drops - though in milli volts or a fraction of it - as one draws current. That means all power supplies have drooping characteristics. This is so because power supply has finite- though very small internal resistance + resistance of all components wires, contacts, switches etc which come in series with the load plays its part in dropping the voltage. This is the reason why full load voltage is less (by few milli volts for an electronically regulated supply) than the no load voltage.

-Bhagirath Mankad

The simplest answer is voltage drop across internal resistance of voltage source.

no load voltage will not have any voltage drops across the loads.

There is a lesson on the subject of regulation linked below. This addresses the effect of gain in the regulator loop. For an even simpler circuit that shows the effect of source impedance for a basic series regulator, click the "Previous" link on the page.

http://www.ko4bb.com/e102/e102-5.php

In this chain of thoughts, it is interesting to see whether it is possible to make the output voltage of a source not only to stay unchanged but even to increase when the load current increases... I think I have seen such a clever trick in the Tietze & Schenk bestseller...

If the load is highly capacitive there are chances that output voltage will be slightly greater than input voltage

The loading current tends to 'pull down' the output voltage, and hence, the full-load voltage is smaller than the no-load voltage.

Alternatively, we can explain the phenomenon by considering  the loading impedance. At full-load condition, the loading impedance is smaller, and therefore, the output voltage is closer to ground.  

CONSERVATION OF ENERGY; Energy can not be created nor destroyed. It is just converted from one form of energy to another. 

Hence, Generated Power = Absorbed Power

Generated Power = VI 

Absorbed Power = Load Power + Copper Losses

==> V_gen*I_gen = V_load*I_load + I_load*I_load*R

As I_load approached Full Load Current then then losses increase and the only value that can change in order to maintain "energy conservation" is V_load. 

It is the same behavior with transformers and batteries. As rated power is approached there is a drop in voltage.

With increase in load the load current passing through the circuit increases and hence the voltage drop across the resistance of the components in series path wrt load increase increases causing a drop in the voltage across the load terminals.

Full load voltage could not be less than no load voltage, in all cases, or regulation could be positive or negative, depending on types of load.

Iff you're talking about a power transformer, then the simplest way to put it (as its mentioned above) is due to resistive losses. The real answer gets more complicated and you have to use the equivalent circuit of the transformer to calculate the actual VR.

Also VR can result in higher Vout for a leading Power Factor at full rated current.