Just to clarify - This error happens on ALL the websites and the IP, there is no difference.
Correct, and this shows us that its a horde issue and not one inside the sjwarts website created by ispconfig. In my post above I referred to the URL inside of horde and not the domain you access horde with. If it would be a general rewrite issue e.g. as if mod_rewrite is not active like the horde devs told you, then nothing in horde would work, not even the horde login. But your problem is just a single function in horde is not working, if I understood you correctly, its just that the task manager does not work and that indicates that something in horde or in a .htaccess file in the horde install folder handles this taskmanager URL differently then the other functions in horde.
Please post the .htaccess file from horde, maybe we find something in there.