Feynman's path integral formulation and faster-than-light paths

The paths of the Feynman path integral are not actually taken. The phrase "takes every possible path" is a mangled statement of the mathematical instruction to take the integral of $\exp(-\mathrm{i}S)$ over all possible paths for the action $S$ to get the probability amplitude of something happening. It is a fact of quantum mechanics that this integral computes the correct quantum mechanical amplitude, but the formalism of quantum mechanics never says anything about the particle "taking" these paths, which is in particular absurd because quantum objects are not point particles that have a well-defined path in the first place. So, well, you can say that it "takes" every possible path as long as you don't literally imagine a point particle zipping along each path. Which is what "taking" a path usually means. Which is why this figure of speech does not actually convey any physical insight.

The physical insight lies in understanding how the path integral reproduces the correct quantum mechanical amplitude, which cannot be done on the level of such crude heuristic statements based on classical notions of "path" and "particle". There is no path a quantum particle takes unless you continually track it, and then you'll get a perfectly ordinary classical path (see, for instance, the perfectly normal paths in bubble chambers, where the continual interaction with the bubble chamber effectively tracks the particle).

I would like to add a few things to ACuriousMind's answer. What Greene most certainly intends to say is every path(even faster than light ones i.e. those which are not time-like everywhere) contributes to the propagation amplitude. In fact,since every path in space-time contributes with equal weight, there are also paths which go "back in time" and "come forward" again. For these paths, at a given time, there are multiple positions for the particle which increases the number of degrees of freedom needed. This shows that the single particle picture is no longer consistent with special relativity and one needs infinite degrees of freedom. i.e. a quantum field theory. If you want to study this, you can follow this lecture series by T. Padmanabhan or you can look into his new book.