Maybe this is due to ignoring the effects of temperature when deriving the Hamiltonian.
How are you defining heat at the quantum level? The Schrodinger equation describes how objects behave at the quantum level and heat describes a form energy which is transferred between objects of different temperatures. Temperature is a macroscopic quantity and not microscopic. The equation above describes the energy of electrons in a specific shell and these electrons can change energy by the absorption or emission of photons, and photons do not possess temperature.
No sources I have seen mention any assumptions about temperature.
For exactly those reasons.
Are electron orbital energies dependent on temperature in spite of this common equation?
No they are not.
If you have hydrogen gas heated to near ionization then it should take less than 13.6 eV to remove an electron.
No. Heating hydrogen will not cause the absorption of photons which is needed to ionise hydrogen (other methods for ionisation of atoms exist, but I’m speaking in the context of this question). Moreover, the hydrogen will be ionised upon the absorption of photon with this energy and not any less. This is the crux of the term energy and other quantities are quantised at a microscopic level which gave rise to quantum mechanics.
Is it 13.6 eV no matter what temperature it is or is there a temperature dependency?
Once again temperature is not relevant here. For ionisation to occur, a photon must be absorbed (there are other ways to ionise atoms too). So to answer your question, there is no such dependence.