How do I remove an HTML element with it's content from a specific WP page?

One way to accomplish this would be to create a specific page template for that page. You could accomplish this by either creating a template and only assigning it to the page you need the div removed from, by using the page slug in the $slug-page.php naming convention, or the ID in the $id-page.php convention. Personally I suggest an assigned template so that by changing the page slug or id the page doesn't break.

Once you have the template created, copy over the code from page.php and simply remove the div from the page template.

I think this would be the simplest way to solve the problem if removing the div from the markup is your goal. The only other way I can think would be to edit the page.php template with a conditional statement that checks for that specific pages id or slug and, when matched, won't run the code that creates that div.

If, though, the div you wish to remove is generated from a function, and not a part of any template markup, then you need to identify the function that is producing that markup and look up what filters, if any, it has. I still say it would be wise to create a template file and put the filter in there and assign it to that page as oppose to using a conditional statement in your functions.php.

As for how a function may look for filtering here's an example of a filter function I wrote just today:

/**
 * Edit submit button markup
 *
 * @since 
 * @param array $defaults Existing attributes
 * @return array Ammended attributes
 */
function genfound_comment_form_submit( $defaults ) {
    $defaults['submit_field'] = '<div class="form-submit row">' . 
                            '<div class="' . 'small-12' . ' ' . 'columns' . '">' .
                            '%1$s %2$s' . 
                            '</div>' .
                            '</div>'
                            ;
    $defaults['class_submit'] .= ' ' . 'button';

    return $defaults;
}
add_filter( 'comment_form_defaults', 'genfound_comment_form_submit' );

basically what's happening here is I create a new function and pass the variable that applies to the filter I plan to hook into. In this case its the comment_form_defaults filter hook. This is something you find via WordPress codex or developer reference. From what I read in the documentation I knew I could use a few things from the array stored in $defaults. In this case I pulled the markup for the wrapper that goes around a comment form submit button, as well as the classes associated with the submit input field. In the first I'm overwriting the string that's stored in that item in the array, whereas in the second I just added to the string by using .= instead of =.

Once you're all done making you're changes to whatever items or variables simply return the variable and then hook it into the filter. Also, you can put the add_filter() function before or after the function you write. I've seen it done both ways much of the time, and in this case things just seemed better organized when I did it this way.