We can't sync this section because you don't have write permissions: OneNote.

Issue: While upgrading from our trail version of SharePoint 2013 POC to a usable set-up SharePoint 2013. We have seen an issue reported where syncing your OneNote with SharePoint resulted in an issue and get the following error.

Error: We're sorry. We can't sync this section because you don't have write permissions. Talk to your administrator to resolve the issue. (Error code: 0xE4010071)"

Troubleshooting:

We tried to download the OneNote from the site and opened it, it worked fine.

Locally there was no issue, all the updated changes were saved.

Saving or synchronizing them with SharePoint was having issue.

Opening one note book from SharePoint site was opening as read-only, despite of user having permissions.

As a part of SharePoint 2010 upgrade while working with previous clients I remember that when a site being upgraded it temporary locks for editions.

Resolution:

Thinking of the same ground, I found the MS evidence http://support.microsoft.com/kb/2777002, which talks about the same behavior.

There is workaround to this issue:

1.    Open the SharePoint site, and then open the library that contains the notebook.

2.    In the Connect & Export area on the LIBRARY tab, click Open with Explorer. 

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3.    Copy the notebook folder that contains the section files to the local computer, and then open the section files by using OneNote.

4.    And then update your information.

Once SharePoint is online you can again synchronize your updates.

If you have any issues, do let me know. I would be happier to answer those.

Reference: http://blogs.msdn.com/b/david_rasmussen/archive/2007/12/10/onenote-shared-notebooks-options-and-troubleshooting-part-1-windows-file-shares.aspx 

Product Applies to: SharePoint Server 2013, One Note 2013.