I think there are 2 main sources of confusion:
First, because of gravity, extending your arms feels like work. We're only interested in the radial movement, though, and in this direction, the skater's arms are pulled by the centrifugal force (in the long tradition of spherical cows in vacuum, we could replace the figure skater with two beads on a spinning rod).
Second, the idea of rotational energy as kinetic energy. The relevant work variable is (as already mentioned) the radial extension of the skater's arms, and as far as that's concerned, rotational energy plays the part of potential energy.
Think of the skater pulling in her arms as compressing a spring, and extending the arms as its release.
Going by either the bead or spring model, the rotational energy gets converted into kinetic energy of the arms, accelerated by the centrifugal force in direction of the radial work variable and ultimately dissipating via vibrations when the arms abruptly reach maximal extension.
Of course, if the skater doesn't let her arms be accelerated and slowly extends them instead, the energy dissipates right away, which might be the more realistic approach.
Best Answer
We know from conservation of angular momentum that $I$$\omega$ = constant. So when the object is heated, the body expands leading to a change in moment of inertia. If the new M.O.I. is $I$2; then as $I$2 > $I$1 , automatically $\omega$1 < $\omega$2. Therefore the problem of decreasing rotational kinetic energy is solved, as work is done to expand the material.
Note: Expansion takes place partly from the heat supplied too. As values are not given we cannot calculate for sure.