Teachers have asked if they can re-order the list of sections that they teach
Yes, with limits.
The order that sections are displayed is determined by Period (each Section has a Period field which is intended for use in Scheduling) and then, if the period is the same, it uses Section start time. If there is no period entered, it uses start time. Other than these two fields, you can not manually change the order. This order is used on the admin view (People page > School tab) and the teacher view (Left side bar and Quick Access menu).
If your school is not using the Period field then you could make your sections display in any order you want by using the Period field to number them in the desired order. The below examples are in alphabetical order because they are numbered one through four in the Period field.
Admin View (People page > School tab)
Student sections are also ordered by period (matches their schedule). As a result, this is also the order that a Guardian will see their children's sections.