Based on our engineers’ further research, we discovered the possible cause for the majority of these cases is installing the OEM version of Office on a machine. Whether the OEM version of Office is ever run or not, an Unlicensed Product error will appear on boot if we install either the Home or OSub version of Office 2013. And back to your current issue, we suggest changing the permissions of the PayloadOverride key in the registry.

Before offering this method, we’d like to point out this method applies only to computers that are running Windows 7. Thus if you're running Office on a different operating system, please feel free to post back.

To change the permissions of the PayloadOverride key in the registry, follow these steps:

1. Close all open Office programs.
2. Open Registry Editor. To do this, follow these steps:
a) Click Start, click All Programs, and then open the Accessories folder.
b) Click Run.
c) Type regedit, and then click OK.
3. Search for PayloadOverride by using Find on the Edit menu, or browse to the following subkey:
4. Expand the items under the PayloadOverride key, and then right-click the SKUID key.
Note If the SKUID key doesn't exist, this method doesn't apply to the computer. Post back to let us know that.
5. Click Permissions.
6. Under Group or user names, select Everyone.
7. Click the Full Control check box under Allow in Permissions for SKUID for the Everyone group.
8. In Permissions for SKUID, click Advanced.
9. In the Advanced Security Settings for SKUID area, click Everyone under Permission entries.
10. Select the Replace all child object permissions with inheritable permissions from this object check box.
11. Click OK.
12. In Windows Security, click Yes.
13. In Permissions for SKUID, click OK.
14. Open or start any Office application.
15. On the File menu, click Account to verify that Office is activated.

See Microsoft KB article for original post.

Here’s a KB article for your reference:
"Unlicensed Product" or error 0x80070005 in Office 2013 or Office 365