Two rectangular parallelepiped

This version works for all parallelepipeds, not only rectangular ones:

If you replace each parallelepiped by all points that have distance at most $\varepsilon$ to a point in the parallelepiped, you can still place the smaller inside the bigger one. In particular, the smaller object has a smaller volume.

We divide up the extended parallepipeds by extending the planes corresponding to the six faces. This gives the volume of the original solid in the center, parallelepipeds of height $\varepsilon$ on top of each face, partial cylinders (with slanted parallel ends) of radius $\varepsilon$ and length the corresponding edge, and partial spheres of radius $\varepsilon$ around each vertex.

The partial spheres add up to exactly one whole sphere simply by translation. The partial cylinders corresponding to parallel edges add up to one whole cylinder by translation.

Now let $\varepsilon$ tend to infinity (yes, really). The term with $\varepsilon^3$ comes from the sphere of radius $\varepsilon$ and does not depend on the parallelepiped at all, . The coefficient of $\varepsilon^2$ comes from the cylinders and is clearly the sum of the edges times some constant.

So, for the inequality to be valid the sum of the edges of the smaller box must be smaller than the sum of the edges of the larger box.

I don't see any problem with the generalization to higher dimension.

This is true for 3-dimensional parallelepipeds. Let the sides of the ambient parallelepiped be $A,B,C$ and of the inner one $a,b,c$. Then it's clear that $A^2+B^2+C^2\ge a^2+b^2+c^2$ since the diameter of the ambient body is obviously bigger. Also, it's easy to see that the there is a 1-Lipschitz map of $\mathbb R^3$ onto the smaller parallelepiped (just take the nearest point projection map). Being 1-Lipschitz this map does not increase 2-dimensional area and therefore, applying this to the surface of the bigger parallellepiped we get $2(AB+BC+AC)\ge 2(ab+bc+ac)$. Adding the above two inequalities we get $(A+B+C)^2\ge (a+b+c)^2$.

I learned of this problem from Anton's Petrunin's list of exercises in orthodox geometry (which has a lot of other cool problems btw).

I don't know if the same holds in higher dimensions.