Principle of locality Why does the principle of locality have so such great importance in physics that theory should be consistent with it?
 A: This is really the same answer as Jamal's, but I wanted to express it in simpler terms.
You may know that spacetime points that are simultaneous in one inertial frame are not necessarily simultaneous in a second inertial frame moving with respect to the first. This causes problems if we allow non-local interactions.
Suppose we have some system $A$ that interacts with some system $B$. If the interaction is local that means it occurs at the same spacetime point for both $A$ and $B$. If we make this point the origin of our coordinate system, $P_A = P_B = (0, 0)$, then however we Lorentz transform we find that in all inertial frames $P_A = P_B$. So if the interaction is local in one frame it's local in all frames and everything is cool.
But suppose we allow a non-local interaction i.e. at the moment of the interaction $P_a \ne P_B$. Let me draw this in the rest frame of $A$ and $B$:

In this frame both points $P_A$ and $P_B$ have $t = 0$, but the spatial positions are different so $A$ is at $x = -a$ and $B$ is at $x = a$. So the interaction happens at the same time for both systems but it's non-local because it doesn't happen at the same place for both systems.
Now look at the interaction time for a frame, $S'$, moving with velocity $v$. The Lorentz transformations tell us:
$$ t' = \gamma \left(t - \frac{vx}{c^2} \right) $$
If we work out the time of the interaction in the moving frame we get:
$$\begin{align}
t'_A &= \gamma\frac{va}{c^2} \\
t'_B &= -\gamma\frac{va}{c^2}
\end{align}$$
So in our moving frame it now looks as if the interaction happens at different times because $t_A > t_B$. But worse is to come. Now look at the interaction from another frame, $S''$, moving at velocity $-v$ (i.e. in the other direction to $S'$). Using the Lorentz transformation again we find the interaction times in $S''$ are:
$$\begin{align}
t''_A &= -\gamma\frac{va}{c^2} \\
t''_B &= \gamma\frac{va}{c^2}
\end{align}$$
So in $S''$ we find $t_A < t_B$ instead of $t_A > t_B$.
So the problem is that if we allow non-local interactions not only does the interaction not look simultaneous in different frames but we can't even define causality because in some frames the interaction at $A$ precedes the interaction at $B$, while in other frames the interaction at $B$ precedes the interaction at $A$.
This problem arises for any spacelike interval, and indeed it's one of the reasons why faster than light travel causes problems, because we get similar breakdowns of causality. The simple way to avoid this is to say that all interactions must be local i.e. the two interacting systems must interact at the same spacetime point.
A: If we abandon locality, a consequence would be that forces would transmit instantaneously at all points, and we know from observation, as well as relativistic arguments that this is unreasonable. In quantum field theory, locality is a required axiom to ensure causality.
To elaborate, for any theory to be causal, we must have for all operators,
$$[\mathcal{O}_1(x),\mathcal{O}_2(y)]=0 \quad \forall (x-y)^2 <0$$
for spacelike $x,y$. It ensures that a measurement or observation at $x$ cannot influence $y$ when the two measurements are not causally connected. Of course, for timelike separation only, it may not hold. For example, for a real scalar field, with $x-y=(t,0,0,0)$, we have,
$$[\phi(x,0),\phi(y,0)] \sim e^{-imt}- e^{+imt}$$
but certainly does vanish for spacelike separation. Issues with causality still remain; there is a general disagreement as to whether it has been established that quantum entanglement either does or does not violate the principle of locality.
A: If we dismiss the principle of locality than we are back to the times of Newton and his force at a distance: a mystical way of transmitting force (causality) between distance-separated bodies. We would be actually in a world of magic and sorcery (only that here it would be science-approved) :)
In other words, the principle of locality makes scientists to show a (falsifiable) way objects interact with each other. This means that a force (cause) must be transmitted somehow to the immediate vicinity of the object it acts upon. Without showing how it is done, you can claim almost anything (and the speed of light would not be a constraint).
A: First: special relativity (and its principles) is a well-established theory, decades of experiments agree with it, so that it is most natural to include it in new theories rather than start from scratch. If something works, why should you reject it? The day some experiment will prove it to be wrong, then we will think about it, not before.
Second: from a more philosophical point of view, there are some principles we try to put in physical theories because allow us to do science. For example, the Laplacian determinism, or unitarity in quantum mechanics, allow us to describe the dynamic of a system without any privileged starting time. This is very useful, because we cannot think of carrying out experiments at the same time, nor we cannot think of study the universe itself by the beginning: we can just study it with experiments done at our time.
Locality is a principle like that, by requiring that distant part of the universe does not influence themselves, we can study it by looking at what is close and accessible to us.
