I played around with this some and was able to accomplish it by doing the following:

1. On the QTP Machine, from Control Panel, go into Administrative Tools and then open Services.
2. Start the WebClient service. You want to set this service to automatically start.
3. Open a Windows Explorer and clik Tools
4. Click on Map Network Drive...
5. Select a drive letter and put the UNC path to your file on SharePoint in the Folder box
6. Select BOTH checkboxes, Reconnect at logon AND Connect using different credentials
7. Click Finish
8. Supply proper credentials to access the file, Select the Remember my credentials checkbox and click OK. It may ask for credentials again.
9. Change the code to access the mapped drive rather than the UNC path to Sharepoint

This method worked for me. It seems that you have to be able to supply credentials to SharePoint to access the files, and also the WebClient Service needs to be started in order for UNC paths to work properly. When you restart the computer, you need to ensure that the mapped drive connects properly before trying to run a QTP test.