How to display data collected in a form on a page templated in module's tpl file?
You need to use hook_theme in your module to manage content in tpl file. 

Example,

Creating template file using the hook_theme function

    /**
    * Implements hook_theme
   */

    function example_theme(){
      return array(     
        'exampletheme' =&gt; array(
            'template' =&gt; 'exampletheme',
        'arguments' =&gt; array('example_array' =&gt; NULL),
          ),
      );
    }


Now, we can use "exampletheme" theme to display content. Below is the page callback function for the menu item.

 function example_callback(){

  $array = array(1,2,3); 

  return theme('exampletheme',  $array);
}


Now, you need to create "exampletheme.tpl.php" file and you can render the "example_array" in the tpl file.