If dark matter (DM) turns out not to be particles it does not break physics so much as it might break physicists. Erik Verlinde posits that DM is due to the role of entanglement in spacetime. For the de Sitter spacetime AdS entanglement with the horizon, or the anti-de Sitter spacetime with its boundary, is contrasted with entanglement between particles in a de Sitter spacetime. Verlinde uses the following diagram to illustrate this:
These entanglements carry dark energy across these Einstein-Rosen (ER) bridges that are defined in the ER = EPR equivalency. This results in different bulk distributions of entanglement entropy, and thus different curvature perturbations of spacetime. For pure entanglement with the boundary, such as what would happen with AdS this is a form of the AdS equivalency with a conformal field theory (CFT) on the boundary. Then gravitation in the bulk of the AdS is equivalent to the CFT on the boundary. For the right hand case in dS these entanglements are "free" and not exclusively attached to a boundary or horizon.
For the Schwarzschild spacetime with two timelike patches a particle inside a black hole can be entangled with a state in the two regions I and II. In the diagram below the red loop corresponds to an $O(\hbar)$ quantum fluctuation or loop can appear to an observer as a particle seen as the red dot in region I that is entangled with a particle in region II. However, since this is superposed with a loop both particles are equivalent to a particle inside the black hole corresponding to any point on the loop in region III. In this way a bipartite entanglement of a particle with a black hole does not turn into a tripartite entanglement after the page time, because there has been ultimately a redundancy in counting particles! These ER bridges identified with entanglements are the region III of black holes, and ER = EPR means that quantum entanglement is fundamentally equivalent to a black hole, and that quantum entanglement in a way "stitches together" regions of spacetime, such as these ER bridges.
Erik Verlinde takes this a bit further to argue there can be a distribution of entanglement entropy in spacetime that has gravitational content, but which is not due to particles. Verlinde then argues this results in a MOND-like form of deviation of spacetime that we observe as dark matter. In a funny sense Verlinde then argues that DM is due to a quantum effect, which is similar to the question here.
The idea is interesting, and it could be the case. However, I will say that I have problems with it. I would argue with black holes, where the remnant of a black hole is not a Planck mass black hole that is somehow stable, but elementary particles. I would argue that while Verlinde has an interesting comparison about entanglements with horizons and boundaries with entanglements in the bulk, I would argue that the entropy of entanglement in the bulk is ultimately due to just elementary particles. I think Verlinde's analysis might possibly be extended to show that this can generate some form of elementary particles.
So I would say that as things stand now, which is with a big range of uncertainty of knowledge, that it could be the case that Verlinde's argument or something similar could demonstrate that DM is a pure quantum effect.