I need to have a background process that runs independent of my app and performs a set of tasks. These tasks need to execute even when my app isn't running. For example, continuously process a list of tasks that contain the date/time they need to be executed on. The background task would iterate over the list and process all of the ones that match the current time.
Is this possible to do locally without the need for a web server and utilizing the push notification services?
Here is a link to an overview of Background Agents on Windows Phone. At this point what you want is not possible with any degree of accuracy (periodic background tasks are run only once every half hour at the minimum) or dependability (background agents are disabled if the user does not open the associated app for a while).
So yes, at this time your only option is to create a push notification server and have that notification deep link to whatever app you want to open.