Append HTML Using Shortcode

First things first. You shouldn’t really use echo inside a shortcode, unless there is NO way to return the value. In your case your can simply convert it to a function that returns a value:

function date_important_info(){

    return display_date($date);

    function display_date($date){ 
        $data = "
            <div id='my-shortcode' style="">
                <span>Date:</span>
                <span> {$date} </span>
            </div>";
        return $data;
     }
}

add_shortcode('date_info','date_important_info');

Now, about your problem with appending the data. You can select the HTML output of your shortcode and append it to another element by using jQuery. Here’s a quick example:

// Get shortcode's content
var content = $('#my-shortcode').outerHTML();
// Add it after the element you want
$('#the-element').after(content);

I’ve added an ID to your shortcode’s wrapper, to be able to select it via jQuery. You can also use .append() or .before(), based on your needs.