I'm using the "assignee" field to assign a user to an item within my DA but it doesn't show up on their timesheet. I only allow my team members to update tasks through their timesheets b/c I want to keep track of the hours they are logging. For example, I'm creating a dynamic app for "Report Requests" so team members can log requests for reports and assign the appropriate person. This works great, except the task doesn't show up on the assignee's time sheet. Is this possible with dynamic applications?

Timesheets only show Tasks from the native Task Application.  In your use of the DA you are calling a line item a Task but because a DA can be anything, the assignee field could mean anything thus it wouldn't be appropriate to display those items in Timesheets.  To accurately track time agains DA items they need to be manually typed in as a Task or:

Create a global tab for your DA and open it up to your timesheet only people so when they login they see Timesheets and the one DA tab.  Be sure to give them an appropriate starting view of that DA.  Create a single task or a non-project work item and have users log their Report Request time against that.