How can I show/hide HTML elements if the page is_front_page

This is really not enough information, so I have to guess.

A) The div is part of the content

Use PHP, check for front-page, and if not, handle the div:

<?php if (! is_front_page()) : ?>
    <div />
<?php endif; ?>

B) The div is located somewhere else on the page

Give the div an ID and use jQuery:

<?php if (! is_front_page()) : ?>
    <script>
        jQuery(function ($) {
            $('#DIV-ID-HERE').remove();
        });
    </script>
<?php endif; ?>

// EDIT
Of course, method A should be preferred! I’d suggest to include the above code where the div is generated.