Wordpress - How can I remove the "Add New" button in my custom post type?

A prettier solution would be to disable the capability of creating a custom_post_type:

Simply pass the the parameter 'create_posts' => 'do_not_allow', in the capabilities array when calling register_post_type.

$args = array(
        'label'               => __( 'Custom Post Type', 'text_domain' ),
        'description'         => __( 'Custom Post Type', 'text_domain' ),
        'labels'              => $labels,
        'supports'            => array( ),
        'hierarchical'        => false,
        'public'              => true,
        'show_ui'             => true,
        'show_in_menu'        => true,
        'show_in_nav_menus'   => true,
        'show_in_admin_bar'   => true,
        'has_archive'         => true,
        'exclude_from_search' => false,
        'publicly_queryable'  => true,
        'map_meta_cap'        => true,
        'capabilities' => array(
            'create_posts' => 'do_not_allow'
        )
    );
    register_post_type( 'custom_post_type', $args );

Please refer below :

function disable_new_posts() {
    // Hide sidebar link
    global $submenu;
    unset($submenu['edit.php?post_type=CUSTOM_POST_TYPE'][10]);

    // Hide link on listing page
    if (isset($_GET['post_type']) && $_GET['post_type'] == 'CUSTOM_POST_TYPE') {
        echo '<style type="text/css">
        #favorite-actions, .add-new-h2, .tablenav { display:none; }
        </style>';
    }
}
add_action('admin_menu', 'disable_new_posts');

Edited @TompaLompa's answer as it was incomplete. Adding create_posts => false won't work if map_meta_cap is not set to true.

Not setting this parameter (or setting it to false) will disable editing of the post type too. This is because you would need to add edit_post capability to each user role in order to add AND edit your post type.

Setting this parameter will use WP internal default meta capability handling and make it work for you if you don't need more finer control over role capabilities then the default WP ones.