Relative post date in recent posts list

You don’t generate a WP Loop using wp_get_recent_posts() so get_the_time() won’t pull the post date. The post date is available within the array wp_get_recent_posts() returns though: $recent['post_date']. Or you can use the modified date: $recent['post_modified'].

This works (tested on my dev site):

<?php
$args = array(
    'post_status' => 'publish', 
    'numberposts' => '30', 
    'tax_query' => array(
        array(
            'taxonomy' => 'post_format',
            'field' => 'slug',
            'terms' => 'post-format-aside',
            'operator' => 'NOT IN'
            ), 
        array(
            'taxonomy' => 'post_format',
            'field' => 'slug',
            'terms' => 'post-format-image',
            'operator' => 'NOT IN'
            )
        ) 
);
$recent_posts = wp_get_recent_posts( $args );
?>
<div class="sidebar-entries">
<?php 
foreach( $recent_posts as $recent ){        
    echo '<div class="sidebar-entry">';
        echo get_the_post_thumbnail( $recent['ID'], 'sidebar-thumb', array( 'class' => 'sidebar-image' ) );
        echo '<div class="sidebar-entries-title">';
            echo '<a href="' . get_permalink( $recent["ID"] ) . '">' . __( $recent["post_title"] ) . '</a>';
            echo '<span class="posted-on">' . human_time_diff( strtotime( $recent['post_date'] ), current_time('timestamp') ) . ' ago </span>';
        echo '</div>';
    echo '</div>';
}
?>
</div>

(A note though, don’t just copy/paste code from the Codex. If you don’t need the taxonomy argument in there, don’t include it.)