No, Windows WebDAV Publishing is not required. Moreover IT Hit WebDAV Server Engine cannot run side-by-side with WebDAV Publishing on the same web application (but they can run together on different web sites on the same machine).
If for any reason you need Windows WebDAV Publishing feature enabled on the web site that runs IT Hit WebDAV Server Engine you can disable the Windows WebDAV Publishing in your web.config:
<configuration>
<system.webServer>
<modules>
<remove name="WebDAVModule" />
</modules>
</system.webServer>
</configuration>