How to rename files located on a network path (UNC path) in Windows 7 command prompt?

Do not use a full path for the second argument. Only the first argument requires a full path. Windows assumes since you are renaming, the file will remain in the same folder as previously specified. It mentions this in the command help at the bottom:

C:\Users\John>ren /?
Renames a file or files.

RENAME [drive:][path]filename1 filename2.
REN [drive:][path]filename1 filename2.

Note that you cannot specify a new drive or path for your destination file.

e.g.:

ren \\servername\folder\file.txt file2.txt

Alternatively you can map a drive letter to the UNC share and then issue a command such as:

ren Z:\file.txt file2.txt