Conservation laws aren't fundamental, but are implications of local differentiable physical symmetries (this is the primary result of Noether's (first) theorem).
The local symmetries in question are represented by the Poincare Group, which is the isometry group of Minkowski spacetime, which in turn is the unremovable background of Special Relativity. (The Lorentz Group is a subgroup of the Poincare Group).
This is another way of saying that in a 3+1 flat spacetime, well-designed local probes of fundamental physics will not depend on time translation (i.e., experimenting today vs experimenting tomorrow), on spatial translation (e.g., the same experimental results here vs there), on spatial orientation (i.e., rotation about the 3 spacelike axes; so you get the same result when you turn the system under test 90 degrees to the left), or under Lorentz boosts, which are basically instantaneous changes in constant uniform motion along any of the axes. Additionally, small-scale (small compared to several times the size and mass-energy of the whole solar system) natural phenomena are essentially always "well-designed local probes of fundamental physics".
Conservation of linear momentum arises from invariance under spacelike translation; a violation of that conservation makes it very difficult to maintain spatial translation symmetry, and in particular flies in the face of the many direct tests of physical systems at different places on our planet and in our solar system, for example. So that's a big deal that needs explaining, and in particular the explanation should preserve the known and reproducible invariance under those translations as well as allowing the EmDrive's alleged violation.
The perpetual motion machine argument is slightly different because it is rooted in a claim about how the EmDrive behaves when in non-constant motion; in the current version of the theory paper "V 9.4" at equations 14-16, there is an implied violation of the Einstein Equivalence Principle. The equation is a bit odd, and you easily can read it to say that there is a power<>thrust relationship that varies with acceleration, and raise the Einstein elevator objection: if you put the EmDrive into an upwards-accelerating box, or leave it on the ground, do you still take this power<>thrust relationship seriously? If so, you get "free power" (although that's a bit subtle). If not, then you have to explain a violation of the Equivalence Principle, which is also something that has also been very well tested and has so far applied without fail.
You could think of it with a somewhat concrete example: to hover at some fixed height above the Earth's surface, the EmDrive would have to emit thrust similar to the amount of work you would do to hold it at that height with your hands; since you are near the surface, that should be about "g" expressed as N/kg (and indeed Eq 16 tells you how much electrical power you will require for the EmDrive). So far so good. But (eq 16) can be read to say that when you place the EmDrive on the ground, it will produce electrical power proportional to to "g".
I think this is pretty clearly just a mistake rather than a serious claim. Unfortunately eq 16 features prominently in the FAQ and all the "marketing" material about the EmDrive. (Even more unfortunately it's hard to see how to fix the equation without making the drive obviously inoperable).