In using OpenErp 7 or Odoo 8, how do I modify it such that a manager assigned to a project is the one who will approve all timesheet entries for it?
You can manage this by grouping each employee according to their privilege. For example you have two groups Managerial and employee group so each of them might have different or some how common privilege on certain python objects from OpenERP so please identify those python objects and explore more in Setting >>  Users >> Groups
I have downloaded and installed and also tested via a virtual machine online Odoo 8 and OpenErp 7. I have spent many hours tinkering with the apps and features of both. I am unable to find any way from hours I spend searching or tinkering for a method to change the approve timesheet functionality in the manner I will explain below.
Each project will have an assigned manager. Any number employees can enter time for a project. Once employees send their timesheet to be be approved, each respective manager will only get that portion of the timesheet for which time was charged to the project they managed. They should be able to view each project and the employees in them.