A comprehensive list of theoretical approximations that are used in computational chemistry
The goal of computational chemistry is to obtain the properties of a system. This is done by solving Dirac's Equations.
Treating particles as point particles with mass
In most computational software, particles are treated as points with some mass. Neutrons and protons may be lumped into a nucleus. However, this is not true in all cases as pointed out in this question: Does computational chemistry include neutrons in ground state calculations? .
Electrons, protons, and neutrons are not simply point particles with some mass and charge. The theory and forces can get quite complex. I haven't been able to find journals that address the effect of this approximation. But I will try to post something if I come across it.
Neglecting Relativistic Effects
Wien2k has a nice summary of relativistic effects that need to be considered [8]. The relativistic effects that need to be included are:

Mass-velocity correction
The Darwin term
Spin-orbit coupling
Indirect relativistic effect

However, relativistic effects usually become important for elements further down the fifth row in the periodic table. This is because relativistic effects are dependent on nuclear charge. The velocity of the electron as well as spin orbit coupling increases as the nuclear charge increases.
This does not mean that relativistic effects do not affect &quot;light&quot; atoms. A good example that shows this is the sodium doublet which is a result of spin-orbit coupling. Nevertheless, solving Schrodinger's equation is enough in many cases.
Most computational chemistry software have the option of scalar relativistic calculations. The scalar relativistic technique was developed by Koelling and Harmon. These are calculations that include the Darwin term and the mass velocity correction. According to Koelling and Harmon's paper:

We present a technique for the reduction of the Dirac
equation which initially omits the spin-orbit interaction (thus keeping spin as a good quantum number), but retains all other relativistic kinematic effects such as mass-velocity, Darwin,and higher order terms.[9]

Some offer fully relativistic calculations, which include all (or most) relativistic effects. But these are rarer to find, and are only available for certain cases.
So why are relativistic effects neglected? Well, relativistic effects are small for light molecules and relativistic calculations are expensive

This is so because relativistic calculations need self-consistent solutions of about twice as many states as non-relativistic ones. [10]

Thus, scalar relativistic calculations offer a nice middle ground between efficiency and accuracy.
The Born-Oppenheimer Approximation
I am not sure what motivated Born and Oppenheimer to use this approximation. The seminal paper is in German.[1] It appears that the motivation was to simplify the Hamiltonian. According to the introductory course at MIT,[2]

it allows one to compute the electronic structure of a molecule
without saying anything about the quantum mechanics of the nuclei

And according to Wikipedia,[3] it reduces the amount of computations:

For example, the benzene molecule consists of 12 nuclei and 42
electrons. The time independent Schrödinger equation, which must be
solved to obtain the energy and wavefunction of this molecule, is a
partial differential eigenvalue equation in 162 variables—the spatial
coordinates of the electrons and the nuclei.

As many approximations, it does not hold true in all scenarios. Cases where the Born-Oppenheimer approximation fail are:[4]

ion-molecule, charge transfer, and other reactions involving obvious
electronic curve crossings

Qualitatively, the Born-Oppenheimer approximation says that the nuclei are so slow moving that we can assume them to be fixed when describing the behavior of electrons. Mathematically(?), the Born-Oppenheimer approximation allows to treat the electrons and protons independently. This does not imply that the nuclei and electrons are independent of each other. In other words, it does not mean that the nuclei are not influenced by the motion of electrons. The nuclei still feel the motion of the electrons. In addition, the Born-Oppenheimer approximation does not say that the nuclei does not move. It only means that when describing the motion of electrons, we assume that the nuclei are fixed.
No Analytical Solution to Dirac/Schrodinger Equation
Unfortunately there is no analytic solution to the Dirac equation for any atom that has more than one electron even after the Born-Oppenheimer approximation (a list of quantum-mechanical problems that have an analytical solution is available on Wikipedia[5]). Many texts state that the reason as to why the Schrödinger equation is not exactly solvable for more than one electron is due to the Coulomb repulsion between electrons.[6]
However, this is not entirely true. A counterargument is Hooke's atom. The Hamiltonian for Hooke's atom has an Coulomb electron-electron repulsion term. However, it has an exact solution for more than one electron under certain circumstatnces.[7]
The true reason as to why the Schrödinger equation is not solvable for multi-electron atoms is due to the fact that the motion of electrons cannot be decoupled from each other. In other words, the Hamiltonian is not separable for a multi electron system. If we were to get rid of the electron-electron Coulomb repulsion, the motion of the electrons can be decoupled. This may be the reason as to why the electron-electron Coulomb repulsion (a.k.a. electron correlation) is used as the reason why the Schrödinger equation is not exactly solvable.
From non-interacting to the real thing
Since the Dirac equation cannot be solved analytically, we must make  models that are solvable and add approximations to it. These approximations are further refined to get more accurate results.
The most simple model (and the foundation for computational chemistry) is the system of non-interacting electrons. As the name suggests, the electrons do not interact with other electrons. This allows us to write the Hamiltonian for all electrons as the sum of one-electron Hamiltonians.
The one electron Hamiltonian consists of a kinetic energy term and a potential energy term.
$$\mathcal{H}=\sum^N_i \left(\frac{\hbar^2}{2m_i\nabla^2 }+V_i\right)$$
The solutions for the non-interacting system of electrons are analytical. And they give a starting point for other calculations. Hartree-Fock, DFT, and solid state physics use this simplified model.
Note: it appears that there are several terms that say something similar to this. There is the independent electron approximation and the central field approximation. I'll dig into literature to see what are the differences between these three terms.
Variations of the non-interacting system of electrons
There are several variations of non-interacting electron systems. In all these, the electrons do not interact with each other. What sets them apart from each other is the potential energy term of the Hamiltonian $V_i$. This potential energy term is often called the effective potential.
Free electron model
This model is used as the starting point in solid state physics. The free electron model describes the behavior of valence electrons in a metal or semiconductor. Here, the potential $V_i$ is equal to 0 for all electrons. It does not feel the effect of other electrons or nuclei. The wavefunctions of this model are plane waves.
Nearly free electron model
The nearly free electron includes a weak periodic potential. In other words $$V_i(r)=V_i(r+k)$$ It is used to describe periodic systems such as ideal crystals. The wavefunctions of the nearly free electron model are Bloch waves. Bloch waves are plane waves that are multiplied by a periodic function.
Hartree method
In the Hartree method, the potential considers repulsion between two electrons and attraction to the nucleus. In this model,

Hartree assumed that each electron moves in the averaged potential of
the electrostatic interactions with surrounding electrons [11]

Hartree replaced the electron-electron interaction with an effective potential that only depended on the coordinates of the $i$th electron. This effective potential describes an electron interacting with an electron cloud.
From one-electron orbitals to the multielectron orbital
As mentioned above, the total Hamiltonian can be approximated as the sum of one-electron Hamiltonians for non-interacting electrons. However, the wavefunction has to satisfy antisymmetry.
Exact exchange
Electron correlation
Some define electron correlation as everything that the Hartree-Fock method leaves out.
Electron correlation: DFT edition
Electron correlation: Post-Hartree-Fock edition
Periodic systems and pseudopotentials

References

M. Born and R. Oppenheimer, Ann. Phys. 1927, 389, 457–484.
doi: 10.1002/andp.19273892002
Born Oppenheimer Approximation. Open Courseware MIT:Introductory Quantum Mechanics. Fall 2005. Section 12 Lecture. (https://ocw.mit.edu/courses/chemistry/5-73-introductory-quantum-mechanics-i-fall-2005/lecture-notes/sec12.pdf)
(https://ocw.mit.edu/courses/chemistry/5-73-introductory-quantum-mechanics-i-fall-2005/)
Born-Oppenheimer approximation. Wikipedia (https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation)
L. J. Butler, Annu. Rev. Phys. Chem. 1998, 49, 125-71.
PMID: 15012427
doi: 10.1146/annurev.physchem.49.1.125
List of quantum-mechanical systems with analytical solutions. Wikipedia
LibreTexts: 9.1: The Schrödinger Equation For Multi-Electron Atoms
Hooke's atom. Wikipedia
Summary of relativistic effects. Wien2k  (http://www.wien2k.at/reg_user/textbooks/WIEN2k_lecture-notes_2013/Relativity-NCM.pdf)
A technique for relativistic spin-polarised calculations. Journal of Physics C: Solid State Physics, Volume 10, Number 16
(http://iopscience.iop.org/article/10.1088/0022-3719/10/16/019/meta)
The Scalar Relativistic Approximation. Takeda, T. Z Physik B (1978) 32: 43. doi:10.1007/BF01322185 (http://link.springer.com/article/10.1007/BF01322185)
Density Functional Theory in Quantum Chemistry. Tsuneda, T. 2014. ISBN: 978-4-431-54824-9. Page 36.