vee

Did you try this in js file?

jQuery(function () {
  jQuery('[data-toggle="tooltip"]').tooltip()
})

I read the Bootstrap doc it said

For performance reasons, the Tooltip and Popover data-apis are opt-in, meaning you must initialize them yourself.

One way to initialize all tooltips on a page would be to select them by their data-toggle attribute:

You should use get_avatar() function to get author’s avatar/gravatar.
https://codex.wordpress.org/Function_Reference/get_avatar
More references for use.

if (!function_exists('bootstrapBasicPostOn')) {
	/**
	 * display post date/time and author
	 *
	 * @return string
	 */
	function bootstrapBasicPostOn()
	{
//		$time_string = '<time class="entry-date published" datetime="%1$s">%2$s</time>';
        $time_string = '';

//		printf(__('<span class="posted-on">Written by %1$s</span><span class="byline"> Written by %2$s</span>', 'bootstrap-basic'),
		printf(__('<span class="byline"> Written by %1$s</span>', 'bootstrap-basic'),
			sprintf('<span class="author vcard"><a href="%1$s" title="%2$s">%3$s</a></span> %4$s',
				esc_url(get_author_posts_url(get_the_author_meta('ID'))),
				esc_attr(sprintf(__('View all posts by %s', 'bootstrap-basic'), get_the_author())),
				esc_html(get_the_author()),
				get_avatar(get_the_author_meta( 'ID' ))
			)
		);
	}// bootstrapBasicPostOn
}