How do I list the_tags() into HTML data-attribute

Use json_encode():

<li class="griditemleft" data-tags="<?php
$posttags = get_the_tags();
$data = array();
foreach($posttags as $tag) {
    $data[] = $tag->name;
}
echo json_encode( $data );

Later, in your JavaScript, iterate over the li items and for each item use:

// get an array of tag names with jQuery's parseJSON()
var itemtags = $.parseJSON( item.data('tags') );

See jQuery parseJSON() for details.

Update

Actually, you don’t even need parseJSON() in your case, it is much easier with simple tag names. Let’s take a simple function to build a list of the last five posts:

add_action( 'wpse_69446_post_list', 'wpse_69446_post_list' );
function wpse_69446_post_list()
{
    $output="";
    $posts  = get_posts();

    if ( ! $posts )
        return;

    foreach ( $posts as $post )
    {
        $tags      = get_the_tags( $post->ID );
        $tag_names = array ();
        $tag_data="";

        if ( $tags )
        {
            foreach ( $tags as $tag )
            {
                $tag_names[] = $tag->name;
            }
            $tag_data = json_encode( $tag_names );
        }

        $output .= sprintf(
            '<li data-tags=\'%1$s\'><a href="https://wordpress.stackexchange.com/questions/69446/%2$s">%3$s</a></li>',
            $tag_data,
            get_permalink( $post->ID ),
            get_the_title( $post->ID )
        );
    }
    print "<ul class="wpse_69446_post_list">$output</ul>";

    // you have to enqueue jQuery first.
    // This is just as oversimplified demonstration.
    add_action( 'wp_footer', 'wpse_69446_js' );
}

Call it in your theme files with

do_action( 'wpse_69446_post_list' );

Now we need the script for the footer:

function wpse_69446_js()
{
    ?>
<script>
jQuery( function($) {
    $('.wpse_69446_post_list li').each( function( i, item ) {
        var li = $(item).find( 'li' );
        console.info( 'item li:', li );
        // an array or undefined if there are no tags
        console.info( 'itemtags:', $(item).data( 'tags' ) );
    });
});
</script>
    <?php
}

The itemtags are an array now or undefined (empty).

Now you can iterate over that array and do something awesome. 🙂