Why can't we screen out magnetism? 'How do magnets work' has been asked often on here. Usually the questioner gets sneered at and referred to 'this question already has an answer'. But keep clicking and you come back, unsatisfied, to where you started. Ultimately it's how does a force act across empty space? At least Richard Feynman had the grace to say he didn't know (https://www.youtube.com/watch?v=wMFPe-DwULM&feature=youtu.be), and maybe it's fundamental, but one theory in QM is exchange particles - either creating 'push' (easier to accept) or 'pull' (harder). The exchange particle for transmitting the electromagnetic force is the photon - often qualified as 'virtual' to deepen the mystery. But photons can be stopped. Why can't the magnetic force be screened out by something that doesn't transmit light (such as the layer of paint between a fridge magnet and the fridge door)?
 A: As you already stated, magnetic and electric fields interact with virtual photons, not real ones (real ones have zero mass and an energy characterized by their frequency, virtual ones are off mass shell). So magnetic fields will interact by exchanging virtual photons with the fields of the atoms and molecules and lattices of the materials where the field lines pass or end.

But photons can be stopped. Why can't the magnetic force be screened out by something that doesn't transmit light (such as the layer of paint between a fridge magnet and the fridge door)?

You are confusing fields with electromagnetic radiation. They are two different physical states. Static magnetic fields are not light, they can be represented with virtual photons at the atomic level, (the quantum framework) Virtual photons do  not have a fixed frequency, and thus cannot build up a classical light wave. The interactions happen locally, quantum mechanically, with spill over fields of atoms and molecules and lattices.
For most materials the magnetic field  lines effectively penetrate through. Mu metal though is a material that shields from magnetic fields because the lines end or turn out of the material, they do not penetrate.

Mu-metal is a nickel–iron soft ferromagnetic alloy with very high permeability, which is used for shielding sensitive electronic equipment against static or low-frequency magnetic fields.

