Is there a somewhat intuitive explanation as to why the exchange of a photon between two particles causes a force between those particles? Is there a difference in the way massless and massive particles act in this regard?
Basically recoil, since photons can also carry momentum. The reason why this recoil can also lead to attraction instead of plain repulsion is that the exchanged virtual photon has a definite momentum, so it can not have a definite position at the same time (the position wave of a virtual photon is stretched out to infinity). The momentum can be transfered from the emitting particle to the receiving particle (then the force is repulsive), or the other way round - it can also be transfered from the receiving particle to the emitting one (then recoil is negative, so the force is attractive). For more detailed information on this topic also see Virtual Particles @ Math.ucr.edu.