What "happens" to the energy of a photon after it is absorbed?

If you are considering a single isolated atom then it's true that the atom has no way of getting rid of the energy from the photon except by emitting another photon. However as soon as the atom is surrounded by other atoms there are various mechanisms for radiationless decay i.e. transferring the energy of the absorbed photon into channels that don't involve reradiating the photon.

In a gas the excited atom or molecule can collide with another atom/molecule and transfer the excitation energy into kinetic energy. This is known as collisional de-excitation (that Wikipedia article is for collisional excitation, but de-excitation is the same process in reverse).

In a solid the energy can be transferred to lattice vibrations, i.e, heat, which is generally known as quenching. In fact in most solids quenching is so efficient that almost no energy is reradiated as photons. Reradiation in fluorescence or phosphorescence is the exception rather than the norm.