When computing the path integral, are paths that go beyond the light cone excluded? If we are dealing with a theory with a Poincare' symmetry that is proper and orthochronous, I would expect that the metric of the path integral would not include any path that breaks this symmetry (like going backwards in time or faster than the speed of light). However, when constructing the path integral, I don't see this being taken into consideration.
Is there something I have left out accidentally? Or do these paths just contribute an absurdly large phase that it gets 'averaged' out? 
In either case, when computing the path integral on a lattice, are such paths excluded to speed up computation time since they don't have an effect on whatever observable you are computing? 
 A: In the path integral approach to QFT, one integrates over all possible field configurations, weighting each configuration with a complex phase proportional to the action of the configuration. This can be expressed in terms of functional integration (https://en.wikipedia.org/wiki/Functional_integration)
$$
\int \mathcal{D}\Phi(t,\vec{x}) e^{iS(\Phi)}
$$
The functional integration capital $\mathcal{D}$ tells you to integrate over all possible values of the field for all points (t,$\vec{x}$) in your domain.
This includes summing over configurations that are all kinds of discontinuous, disconnected, non-causal random fluctuations. However, in the classical limit, the only configurations that significantly contribute to the answer are the ones where the action is stationary. 
A: In quantum mechanics, remember for an action $S$, the path integral is given by,
$$\int\mathcal D[x(t)] \, \exp i S$$
and one sums over all possible paths $x(t)$, which may satisfy certain conditions, such as for example that one has the same endpoints, $x(t_a) = x_a$ and $x(t_b) = x_b$. In the same spirit, in field theory,
$$Z = \int \mathcal D [\phi(x)] \, \exp iS$$
where we sum over all field configurations $\phi(x)$. Remember at in the quantum mechanics case, we summed over any function between those endpoints and so in field theory, we also consider a field $\phi$ which has any sort of value at points in space-time, $x = (t,\vec x)$. 
Notice, without natural units, the integrand is $\exp \frac{i}{\hbar}S$ and in the $\hbar\to 0$ limit is highly oscillatory with the dominant contributions coming from configurations for which $\delta S = 0$. Thus, although we consider a sum over all the paths, the greatest contributions are those which are actually sensible, i.e. satisfy the classical equations of motion, derived from $\delta S = 0$.
For more information on these mathematical ideas, I would suggest reading about the method of steepest descent in Bender's Advanced Mathematical Methods for Engineers and Physicists.
 An unparalleled discussion of path integrals can be found in Path Integrals in Quantum Mechanics, Polymer Physics and Financial Markets which features very explicit calculations.
