Temperature affects a system through two ways: (1) by (loosely speaking) making the nuclei move faster, and (2) by exciting some electrons to virtual orbitals (or equivalently speaking, exciting some molecules from the electronic ground state to electronic excited states). Finite temperature DFT calculations only capture the latter effect, and unless the temperature is very high (say, >2000 K) or the material is a metal, the latter effect is often negligible. A large variety of methods exist for describing the former effect, which can be classified based on whether they approximate the potential energy surface as harmonic, or whether they use molecular dynamics, etc. Enumerating all such methods is beyond the scope of this answer.
For fluorescence, some of the top contributions from temperature include:
- Temperature can change the aggregation behavior. Suppose that the fluorescent molecule can dimerize, and the dimer is not fluorescent, or fluoresces at a different wavelength, or fluoresces with a different quantum yield. When the temperature increases, the dimer gradually dissociates, and the contribution from the monomer fluorescence increases. This is probably the biggest contribution to the temperature effect for most fluorescent molecules, but is one of the easiest to overlook in computations; in fact, a lot of time will be spent on searching for and optimizing the low-lying conformers of the dimer, and it is not trivial (but possible) to calculate the fluorescence quantum yields of all the conformers, in case such a calculation is indeed necessary.
- Temperature can change the Boltzmann ratio of different conformers (or different protonation states, etc.) of the fluorescent molecule. Again, this has to be studied by a conformational search, followed by Gibbs free energy calculations.
- Temperature can change the relative population of the different vibrational energy levels of the electronic excited state. When the temperature is higher, more molecules fluoresce from high-lying vibrational energy levels, giving different shapes of the fluorescence spectrum. In ORCA this can be done by doing a vibrationally resolved fluorescence calculation, whereby the temperature is specified in the %esd input block.
Finally, in case the electronic temperature does have a significant impact on the fluorescence spectrum, incorporating this effect in a TDDFT calculation is highly non-trivial. For example, the TDDFT module of ORCA (and quite a few other common quantum chemistry programs as well) cannot do this, because it performs the TDDFT calculation by solving the Casida equation, which assumes that all orbitals have integer occupation numbers, and this is not true when the electronic temperature is non-negligible. There are a few ways to do finite (electronic) temperature TDDFT, e.g. by Nakai et al.