The IR signature can still be detected because the cool air does not inhibit the radiation. Assuming the second plastic cube was not opaque, then the IR would pass through mostly undisturbed.
In order to completely erase one's IR footprint one would have to have a "cube" that is conductive, as conductive materials impede radiation: something like copper foil, aluminum foil, mylar etc. However, this shielding would only prevent an IR camera from being able to discriminate objects that are obscured by the foil; it would not hide the fact that the foil is at a temperature warmer than normal(if in fact it is). A second layer of foil (spaced apart from the first) would need to be employed.
Eventually, the heat from the first cube would reach the second via convection and so an IR camera could potentially detect that the outer shield is at above average temperature. Circulating cool air between shields would prevent the outside shield from ever becoming warm enough to be detected.
The first shield will not re-radiate IR (such is supposed to be the case with mylar) but will still convect heat, which may still result in an IR footprint, depending on the setup.
So, one shield of foil is fine to conceal visual information but it will not prevent the shield itself from appearing warmer than it should, which for some people may be a concern. A second shield with cooling air running between the two shields is necessary to both conceal discriminating features and the actual temperature of the room.
Since the heat never convects (because it is cooled), neither convection nor (much)radiation makes it to the second sheet of mylar.
For those wanting to protect themselves from rogue IR camera fiends, the aforementioned setup is unrealistic but ideal.