Is the energy of an orbital dependent on temperature?
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.
In addition to @Dr jh's answer, I would like to add that the effect of temperature on the spectra of atoms is part of what is known as as Doppler broadening of the lines.
In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different Doppler shifts, the cumulative effect of which is the line broadening. This resulting line profile is known as a Doppler profile. A particular case is the thermal Doppler broadening due to the thermal motion of the particles. Then, the broadening depends only on the frequency of the spectral line, the mass of the emitting particles, and their temperature, and therefore can be used for inferring the temperature of an emitting body.
I think this question belies a misunderstanding of the nature of temperature.
Temperature is not an input to fundamental physical laws, it is something that comes out of the physical laws when applied to a large number of objects. There is no explicit temperature in the Schrödinger equation for a hydrogen atom. The concept of temperature only emerges when you consider the Schrödinger equation for a large number of hydrogen atoms that can exchange energy with each other.
The Schrödinger equation (or Newton's laws or Maxwell's equations) establish the ground rules for how atoms and molecules (or any other objects) behave. However, it's usually difficult to work out what these rules predict for anything more than simple systems. The point of statistical mechanics is to predict the average behavior of a large number of such simple systems under the constraints provided by the ground rules (fundamental physical laws). Temperature is a concept that emerges from the statistics of large number of degrees of freedom that can exchange energy. Again, it is not an input to the Schrödinger equation or Newton's laws, but a consequence of them!