The biggest factor is the geometry resolution (lowres/highres/..) and the kind of setup the rigger chooses Hopefully always balancing quality and speed. This is very project and people dependent. So I'm not going into that here. This post should just be a small list of things that always work and are not that well known (?):
- Remove vertices from the deformer, if they are not affected, with the "Edit Deformers > Edit Membership Tool". I almost always use this for lattice, wrap, wire, sculpt, .. There are probably cases for all deformers. This can also be achieved by only selecting the affected vertices when creating the deformer.
- Use few skinClusters by merging geometries. This should be used carefully of course, because it can be annoying to skin overlapping geometry (maybe skin multiple geometries and copy skin weights on the final merged mesh when publishing the rig) and other steps in the pipeline have to be considered as well.
- Remove unused influences from skinCluster. Use "Skin > Edit Smooth Skin > Prune Small Weights" before. Each influence costs a little bit of performance, so if you have multiple geometries that are skinned to all skinning-joints this can make quite a difference.
- Connections instead of constraints. A lot of point/orient/parent/scale-constraints can be replaced by simple translate/rotate/scale connections, if the hierarchy is setup in the right way (Maybe by creating an extra null-group to have the same local transformation values). BlendColor nodes can be used for blending. I think scaleConstraints are always the worse option, and can be avoided completely.
- Don't use clusters to deform curves. Instead use a skinCluster (use component editor to tweak skinning weights) or [Credit to Francisco Naranjo:] connect locatorShape1.worldPosition into the curveShape1.contolPoints[x] attribute
- Delete unused Orig shapes. This does not effect the rig speed when interacting, but is about filesize, which increases file-opening / reference-loading time. Whenever you create a deformer, maya also creates an Orig shape (intermediate object). If you duplicate a geometry that had an orig shape, it will get copied as well, but ignored for future deformers. So it basically just increases the filesize with no benefits. You have to remember to delete it, or delete the history before you duplicate. (An easy shortcut-combination to select the orig shape: Arrow down (child), arrow left (latest child)