That bug you linked to is marked fixed and the problem I've found is defiantly not fixed. I think the bug may be related to renaming inside a briefcase folder I own while what I have noticed is related to folders that have been shared with me where I have been given admin rights.
I've installed TC 7.56a on Win7 with webDAV plugin 2.1. Have you altered any settings from the standard install ? Using TC I can't even rename files/folders in my own briefcases let alone those shared with me.
Using Cyberduck on Win7 give me the same HTTP/1.1 424 Failed
CrossFTP on Lucid32 gives me :
This problem exists when using various clients on both Win7 and Linux which says to me that this is a bug in Zimbra.
[R1] RENAME /dav/my.username/test's Shared with user/sub-folder/sub-folder2/test doc.odt to /dav/my.username/test's Shared with user/sub-folder/sub-folder2/test doc1.odt
[R1] Failed Dependency (424)