When an object is heated, energy is absorbed by vibration, translation, and rotation of molecules. How does this heating apply to electrons?
Heating applies to the whole system (nuclei and electrons). However, due to the large difference of mass between nuclei and electrons (larger than 1800), and taking into account that the correct description of the dynamics at atomic level is provided by quantum mechanics (QM), the way a gain of total energy is distributed among different internal degrees of freedom is quite complex.
If translations, rotations and vibrations of molecules could be well approximated by the classical limit of QM, equipartition theorem ensures that at equilibrium the average energy associated with these motions would be the same for each individual degree of freedom. Already for more complex kind of motions (diffusion, motion of defects) things are more complicated, even at level of classical mechanics.
In the case of electrons, but, in some regime also for vibrations and rotations, QM provides the only correct description of the system. Quantum mechanics allows some sharing of the increment of energy also among quantum degrees of freedom. However in a more constrained way than classical mechanics. Details are system dependent. Just to give an example, in the case of a simple metal like the alkalis, part of the electrons closely follow the ionic motion, thus their presence can be absorbed into an increase of the effective ionic mass. The remaining ones (valence electrons) behave very closely like an ideal Fermi gas: most of them are inert to an increase of energy, but those whose energy is close to the Fermi energy by a few $k_BT$.
Electrons do play a key role in how heat energy flows across a material, especially the free and mobile electrons in metals where they contributes to the high heat conductivity. When a body without free electrons is heated, the heat energy is transferred from atoms to atoms primarily by mere vibration of the particles within their sphere of freedom, and the intensity of the vibration depends on how much energy is absorbed. The heating also causes the electrons in orbit around nucleuses that gained heat energy to rise to an higher energy level, and as the atoms cool, these electrons return to their ground state, releasing photons in the process in the form of infrared radiation off the material.
In materials with free electrons zooming around, the story is taken a few steps further. The transfer of energy is enhanced and accelerated by these free electrons gaining energy and transferring it farther than just atomic vibration would because there have a limitless sphere of freedom they can explore in the lattice, so they actually transport the energy. Electrons also tend to move away from a region of higher energy to a region of lower energy (like virtually all things do), and this helps orient their direction of motion towards the colder region of the material, thus speeding up the heat transfer.
In a solid there exist collective vibrations called phonons. There are acoustic phonons in which atoms move as a whole so the density varies. Then there are optical phonons in which valence electrons move in opposite direction to their ions. One can distinguish longitudinal and transverse optical phonons. For reasonable temperatures, that is below melting, it is the phonons that are involved in heating of the solid. Single atomic excitations are relevant to a much higher temperature range.