Fatal error: Call to undefined function menu_overview_form()

The menu_overview_form function is part of the core "menu" module. Even more odd is that nodehierarchy_menu_overview_form (nodehierarchy's hook into menu_overview_form) checks to see if the "menu" module is enabled, and you shouldn't have been able to get to that point unless it was. Have you had any failed updates or other weirdness that may have screwed up your core modules? At the very least, I would try downloading a fresh copy of Drupal 6.20 and replace your /modules directory with a fresh one. It'd probably be good to do a refresh of all your core Drupal files.

Can you try disabling the other modules and seeing if node hierarchy still throws the error?

The conflict was with Menu Access. Both modules are trying to override the menu edit page (admin/build/menu-customize/xxx) which means that the core menu.admin.inc file isn't being loaded automatically anymore. I fixed this issue, but whatever modifications Menu Access is trying to make are being overridden by Node Hierarchy. That's unfortunately the nature of trying to modify core behavior where there is no real API to do so. I hope this doesn't make Menu Access unusable for you.

Same problem with 7.x-2.x-dev version and Drupal 7.x fresh install: minimal install profile + Views + CTools + NodeHierarchy.
Can you port this patch to 7.x?

Issue solved when I enable "Menu" module, so you must add dependency of this module.