It depends. If the energy is high enough, the electron will fly off with excess kinetic energy. Otherwise, excess energy is emitted as photons, either immediately, or through a thermal decay process when the electron is promoted to one of a multitude of (quantized) states slightly higher in energy than the ground state of a particular orbital.

If the Hydrogen is in a metallic state, then there is no meaningful quantization in a certain band, so energy can rapidly be dumped into this band, then will be lost as heat when the electrons settle into the lowest levels of the band.