It really varies.
If the rescue has a facility (a private shelter), then they'll always need volunteers at the shelter -- cleaning kennels, feeding dogs, walking, socializing, etc. They'll also need volunteers to answer phones, and with some experience, help adopters who stop by to visit.
If the rescue doesn't have a facility and keeps dogs in foster care, the type of volunteer they'll likely need the most is foster homes. No rescue ever seems to have enough of those, and each new foster home means more dogs saved. If you can't foster, then they'll likely need help setting up and manning booths at meet-and-greet events, and possibly handling dogs at those events. They may also have some clerical work.
Be creative about what you can offer! We recently had a new volunteer offer to transport dogs to vet appointments for foster-families who work and can't take off to get the dogs across town to their vet appointments -- it was a HUGE help.