I want to stop all the unnecessary SQL services but I'm not sure whether SharePoint requires SSRS, SSAS, and/or SSIS.

SharePoint requires SSRS if you have any reports/webparts etc that use SSRS reports, so if you shut the SSRS service down, the reports/webparts will error out. Similarly if you have any KPIs/Score Cards built in PerformancePoint Services that are created on top off SSAS cubes, they won't work. SharePoint is not directly related to SSIS but make sure you that there is nothing dependent on SSIS before you shut it down. Hope that helps.