The warm box will of course »heavier« because it contains additional energy. Fundamentally there is no mass, it is just a convenient way to talk about the energy hidden in the structure of matter.
Note that I wrote structure and not internal structure and I did so intentionally. As far as we know elementary fermions - quarks and laptons - have no internal structure but - and I am totally not a physicist - there are the Higgs mechanism, the seesaw mechanism, selfinteraction due to vacuum polarization, and other things I have never heard of. As far as I know, this is still an area of active research with unresolved questions, but I think it is also generally believed among physicists that all mass has a dynamical origin.
This isn't quite the same thing as what adds mass to say hadrons like who you're replying to is, but at least according to the Standard Model, leptons as well as quarks start out massless and acquire an inertial mass due to interaction with the Higgs field. You wouldn't say it is "internal structure" but it arises due to interactions rather than being an a-priori property.
