You may find that you're ready to take on new tasks or more responsibility. This really shouldn't be a problem. Many organizations recognise that most potential volunteers today are not available during 'peak hours' on week days, and they have adjusted their schedules to take this into account. So, you should be able to find an interesting volunteer assignment that allows you to work after school, in the evenings or on weekends. Yes, this certainly could happen.
Some organizations do have age restrictions on some of their volunteer positions.
Don't be turned off if you come up against this situation. Ask about the rationale behind the policy. It may, for instance, come from a requirement for their liability insurance. If you don't find the reason convincing, you may want to try to persuade the organization to consider you for a volunteer position you're really keen about.