The nonlinearity of gravity is obvious even in the solar system - it was discovered because Mercury's orbit, when described with the linear theory of Newton's universal gravitation, behaved as if there were an additional hidden mass between its orbit's periastron point and the sun. Or, if you prefer, summing all the sources of gravitation in the solar system is insufficient for describing all the observed orbits in the solar system. (Indeed, even if you summed all the sources of gravity in the galaxy, you wouldn't get the evolution of the relatively eccentric elliptical orbit of Mercury right.)

Precision measurements by satellites around Earth and spacecraft scattered around the solar system reveal the nonlinearity of gravitation, as do precision measurements of systems like Hulse-Taylor and PSR J2222-0137.

A linear superposition law for gravitation is consequently unavailable - again, this can be seen in the solar system, where a linear superposition model helped find a real mass (Neptune, where ultimately the telescope targetting was driven by Urbain Le Varrier's detailed study of the orbit of Uranus in the 1840s, assuming the validity of Newtonian gravitation), but misled astronomers into decades of futile searches for "Vulcan", a hypothetical body inside Mercury's orbit.

There are a bunch of ways one can capture the nonlinearity of observed gravitation. A nice slogan: gravitation self-gravitates. A nice theoretical framework: Newton-Cartan gravity is great for exploring the failure of linear gravity. An easier theoretical approach: the relativistic two-body effective radial potential energy <https://en.wikipedia.org/wiki/Two-body_problem_in_general_re...> or less encyclopedically <https://spaceengine.org/articles/the-anomalous-advance-of-th...> where you can see the same effective potential term written slightly differently, with more about Mercury's orbit.

Quoting the latter:

  Again I don't want to get lost in math, but it's worthwhile
  just to look briefly at what the math is saying here.
  Notice this still has the exact same two terms from the
  Newtonian effective potential: an attraction that goes as
  -1/r, and a repulsion that goes as +1/r^2.  But a new term 
  is added: another attractive term that goes as -1/r^3. This
  means that at very small radii, the -1/r^3 term dominates, 
  and gravitation becomes attractive again, dominating even 
  over the centrifugal effect of your orbital velocity.  

The increased attraction mimics an additional mass in a linear theory.

No linear theory of gravitation is viable for known planetary and astrophysics. At best one can come up with a quasi-linear theory.

This is calculationally unfortunate: solving the full nonlinear Einstein Field Equations exactly is fiendishly hard. Where one can use a linear approximation, essentially every physicist would choose to do so. Indeed, Einstein invented linearized gravity (and various other approximation techniques). Unfortunately, linear theories of gravity can only ever be approximate, as they fail to deal with multibody systems, systems where orbital velocities are even only thousands of kilometres per second, where one wants to trace out radiation (including gravitational radiation), and so forth. Convincing proof of incompatibility between any possible linear theories of gravity and numerous observed physical orbits have been known since the 1960s. Roman Sexl did some really interesting work (alone and with collaborators like Otto Nachtmann) in that area in that decade.

These days one can turn to box 7.1 (sec. H) of Misner Thorne & Wheeler as a textbook starting point.

None of this really has anything to do with quantum mechanics, except with respect to a correspondence principle (e.g. Fraunhofer-like spectral lines have their origin in quantum mechanics, and we can see how gravity rather than just motion affects them).