Assuming that the atoms are stable, is there a formula to determine how much heat energy is required to fully ionize an atom, given the number of electrons, or electron shells, or something else? Would it make a difference how many atoms are being ionized?
Your question confuses two physics frameworks, the classical thermodynamic one, where the variable "heat" is defined, and the quantum mechanical where atoms with their nuclei and electrons are modeled.
Heat may be defined as energy in transit from a high temperature object to a lower temperature object. An object does not possess "heat"; the appropriate term for the microscopic energy in an object is internal energy. The internal energy may be increased by transferring energy to the object from a higher temperature (hotter) object - this is properly called heating.
In the classical framework temperature is related to the average kinetic energy of the atoms and molecules composing a substance under study:
Average means that there is a distribution of kinetic energies with which the atoms and molecules move bouncing against each other:
Here is how higher temperatures have higher fractions i.e. number of molecules at high kinetic energies.
This graph shows the relationship between the average kinetic energy of molecules [½(mass)(velocity)2] at two different temperatures.
The y axis is the fraction of molecules with that kinetic energy. Emin is the temperature at which the molecules of a liquid contain enough kinetic energy to change phase, from a liquid to a gas.
For higher temperatures there will be an Emin for a change of a gas into a plasma , the energy of collisions being enough to separate electrons from the nuclei.
There will always be enough energy in the tail of the distribution for some collisions to be able to ionize an atom or molecule. The energy is the difference in the energy levels to the ionization level, as seen here (second page) for hydrogen.
The temperature at which a specific gas will ionize into a plasma depends on the ionization energy and statistically the phase transition to plasma ( i.e. ionized gas) depends on the substance. Qualitatively phase transitions versus temperature and pressure are seen here:
The plasma phase is important in studies of astronomy.
To summarize , heat is a thermodynamic quantity and is only statistically connected to temperature , which is statistically connected with kinetic energy, and it is kinetic energy distributions which will show if there are enough atoms/molecules with kinetic energy equal or larger than the ionization energy ( a quantum physics quantity) to make a difference. There will always be at a given temperature some molecules with enough kinetic energy to ionize some atoms, but this is in a tail of a kinetic energy distribution. (Do not forget that a mole contains about 10^23 molecules).
Assuming you are talking about the ionisation that results from heating a gas, then this is described by the Saha equation. This is widely used in astrophysics to describe hot plasmas.
One of the inputs to the Saha equation is the ionisation energies of the atoms or molecules involved, but only the ionisation energies are needed and not the full details of the electronic structure. Though obviously the ionisation energies are related to the atomic structure.
To add to the existing answers, and to address the specific concerns you phrase as
how much [...] energy is required to fully ionize an atom, given the number of electrons, or electron shells, or something else?,
then if you know the ionization energy of the given species, as John Rennie points out, the thermodynamics of the Saha equation will tell you how much ionization you'll have at a given temperature.
If you don't know the ionization energy, though, and your starting point is a description of the atom in question with just the number of protons and electrons it contains, then you're going to have a hard time calculating things from scratch. In principle, one can compute the ionization energy by using quantum chemistry methods, but for large atoms this gets relatively complicated and it involves some serious number-crunching rather than just some simple formula - and, even then, the results are typically only considered valid once they've been validated against experiment.
So the real answer is: you go to the lab and you measure the ionization energy.