What could increase the rate of diffusion across a cell membrane?
Several things could increase the rate of diffusion across a cell membrane.
Diffusion is one of the transport processes that doesn't require energy, because it's driven by environmental conditions rather than a specific cell component. Diffusion can also be defined in terms of entropy; organized systems tend to decay without an influx of energy to maintain their organization. Diffusion is the motion of particles from a region of high concentration to low concentration, so all we need to do to affect this motion is alter either or both of the concentrations, or change the particle being diffused.
If, for example, we choose to talk about the concentrations of ions, specifically sodium, then we can look at how pumps and gated channels would affect diffusion. Assuming that the cell membrane is impermeable to these ions, then the only way to let them into the cell is to create channels that allow them to flow in until they reach equilibrium with the outside of the cell; after that point, we have to spend energy using pumps to increase the concentration, and we also need gated channels to prevent them from flowing out until we want them to. So, if the cell has gated channels and pumps, this can control the rate of diffusion and increase it. It can also be further increased by adding additional pumps (to increase the high concentration) or additional gates (to allow particles to move through the membrane faster).
The particles themselves could be changed as well; typically smaller particles will be able to diffuse faster simply by being less difficult to fit through the channels and pumps. You could also change the charge or polarity of the particle, although this would need to be evaluated on a case-by-case basis as the charge might be integral to the function of the gates and pumps.