Yes, it generalizes.
For any two (nondegenerate) nondegenerate tetrahedra A and B, you can find an affine transformation such that T(A)=B. Since affine transformations preserve ratios of line segments and areas and volumes, with a Routh's-theorem-type construction the ratio of volumes volume of outer : original tetrahedron to volume of inner tetrahedra polyhedron will be preserved.
As a commenter pointed out, with a tetrahedron there may be more than one way of defining a sensible cut. Once you've defined this, though, you can find a convenient tetrahedron (maybe a regular one, maybe one with a lot of right angles) and use it to calculate the proportion you want.