This is one funny problem.
It seems that Joomla includes code from module template overrides in a manner that certain variable names do affect the core JModuleHelper renderModule function.
All you need to do to reproduce this bug-like behavior, is to write the following line in your module template override:
$template = JFactory::getApplication()->getTemplate(true);
By loading any object in a variable called
$template, you break your site.
So, the problem has a simple solution. Namely:
Search the module template override code (inside
or possibly even in the module itself (inside
$template and replace with anything else, like
$my_template and this will be fixed.
Make sure to let the developer know, if this is not something written by you.