First of all, there are two different notions of linearity (or superposition) involved in quantum mechanics: at the level of states, and at the level of fields.
In quantum field theory, you have a field ##\psi## (such as the electromagnetic field, or the electron field, or the Higgs field, etc.). This field (in the Heisenberg picture) obeys Heisenberg equations of motion, which looks a lot like the classical equations for fields. For example, a free spin-zero field obeys the EOM:
##(\frac{\partial^2}{\partial t^2} - \nabla^2) \psi = -m^2 \psi##
(You have to stick in ##c## and ##\hbar## in various places to make the units work out.)
This is a linear equation of motion, in the sense that if ##\psi_1## and ##\psi_2## are two solutions, then so is the superpostion, ##\psi_1 + \psi_2##.
If there is a self-interaction, then this leads to a more complicated equation of motion, maybe something like:
##(\frac{\partial^2}{\partial t^2} - \nabla^2) \psi = -m^2 \psi -\lambda \psi^3##
This equation of motion is nonlinear, and does not obey the superposition principle.
Now, there is also a quantum-mechanical notion of state, ##|\Psi\rangle##. In the Schrödinger picture, the state also obeys an equation of motion, something along the lines of:
##H |\Psi\rangle = i \frac{\partial}{\partial t} |\Psi\rangle##
where ##H## involves field operators such as ##\psi## above.
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Rules of Quantum Mechanics require that the equation of motion for the state is always linear, but the equations of motions for fields is not necessarily linear.
So in the case of electromagnetism, the issue is whether the equation of motion for the field are linear, or not. Classically, in the absence of charges, the electromagnetic field obeys linear equation of motion. That means that the electromagnetic field has no self-interaction.
Quantum-mechanically, it's a little more complicated. In the perturbation expansion for the electromagnetic field, there are Feynman diagrams that involve indirect interactions between two photons. The diagram can be loosely described as: "one photon produces a virtual electron-positron pair, and then the electron or positron interacts with the other photon before annihilating into a photon again". Looking at just one such diagram, there appears to be a photon-photon interaction. I am not an expert at quantum electrodynamics enough to say whether this is a real effect. Each diagram has no actual physical meaning, but only represents one term in an infinite sum describing the interaction. I don't know whether the apparent interaction persists when you sum the diagrams, or not.
But assuming that it does, then that means that light can interact with light, and such a self-interaction means that the superposition principle doesn't hold precisely (for fields).
