Use AJAX in shortcode

Question

First off, this is very borderline within the scope of WPSE, it at all.
Apart from the shortcode to trigger the initial HTML output, this is really just AJAX.

Anyhow, that being said, this is how it’s done:

The PHP

Assuming that the above PHP snippet you supplied is functional, place the following in a php file for the ajax call:

/wp-content/themes/%your_theme%/js/ajax-load-quote.php

 <?php
 /* uncomment the below, if you want to use native WP functions in this file */
// require_once('../../../../wp-load.php');

 $array = file( $_POST['file_path'] ); // file path in $_POST, as from the js
 $r = rand( 0, count($array) - 1 );

 return '<p>' . $array[$r] . '</p>';
 ?>

For future reference and so as to make this answer useful to others: Note that wp-load.php must be included in order to make use of native WordPress functionality. The most commom case likely being either the need for WP_Query or $wpdb.

The HTML Structure

In a page content, a widget or a template file:

<div id="randomquotes">
    <p>I would rather have my ignorance than another man’s knowledge,
       because I have so much more of it.<br />
       -- Mark Twain, American author & Playwright</p>
</div>
<a id="newquote" class="button" href="#" title="Gimme a new one!">New Quote</a>

This you can obviously adjust to your liking, but for the sake of this example, this is what we’re going with.
We will generate the above via a shortcode later.

The jQuery

/wp-content/themes/%your_theme%/js/ajax-load-quote.js

function ajaxQuote() {
    var theQuote = jQuery.ajax({
        type: 'POST',
        url: ajaxParams.themeURI+'js/ajax-load-quote.php',
        /* supplying the file path to the ajax loaded php as a $_POST variable */
        data: { file_path: ajaxParams.filePath },
        beforeSend: function() {
            ajaxLoadingScreen(true,'#randomquotes');
        },
        success: function(data) {
            jQuery('#randomquotes').find('p').remove();
            jQuery('#randomquotes').prepend(data);
        },
        complete: function() {
            ajaxLoadingScreen(false,'#randomquotes');
        }
    });
    return theQuote;
}
/* Loading screen to be displayed during the process, optional */
function ajaxLoadingScreen(switchOn,element) {
    /* show loading screen */
    if (switchOn) {
        jQuery(''+element).css({
            'position': 'relative'
        });
        var appendHTML = '<div class="ajax-loading-screen appended">
            <img src="'+ajaxParams.themeURI+'images/ajax-loader.gif"
                alt="Loading ..." width="16" height="16" /></div>';
        if( jQuery(''+element).children('.ajax-loading-screen').length === 0 ) {
            jQuery(''+element).append(appendHTML);
        }
        jQuery(''+element).children('.ajax-loading-screen').first().css({
            'display': 'block',
            'visibility': 'visible',
            'filter': 'alpha(opacity=100)',
            '-ms-filter': '"progid:DXImageTransform.Microsoft.Alpha(Opacity=100)"',
            'opacity': '1'
        });
    } else {
        /* hide the loading screen */
        jQuery(''+element).children('.ajax-loading-screen').css({
            'display': '',
            'visibility': '',
            'filter': '',
            '-ms-filter': '',
            'opacity': ''
        });
        jQuery(''+element).css({
            'position': ''
        });
    }
}
/* triggering the above via the click event */
jQuery('#newquotes').click( function() {
    var theQuote = ajaxQuote();
    return false;
});

Putting it together in functions.php

Below your above snippet (which you will find included modified below), paste the following:

function random_quote( $atts ) {
    /* extracts the value of shortcode argument path */
    extract( shortcode_atts( array(
        'path' => get_template_directory_uri() . '/quotes.txt' // default, if not set
    ), $atts ) );
    $array = file( $path );
    $r = rand( 0, count($array) - 1 );
    $output="<div id="randomquotes">" .
            '<p>' . $array[$r] . '</p>' .
        '</div>' .
        '<a id="newquote" class="button" href="#" title="Gimme a new one!">New Quote</a>';
    /* enqueue the below registered script, if needed */
    wp_enqueue_script( 'ajax-quote' );
    /* supplying the file path to the script */
    wp_localize_script(
        'ajax-quote',
        'ajaxParams',
        array(
            'filePath' => $path,
            'themeURI' => get_template_directory_uri() . "https://wordpress.stackexchange.com/"
        )
    );
    return $output;
}
add_shortcode( 'randomquotes', 'random_quote');
/* register the js */
function wpse72974_load_scripts() {
    if ( ! is_admin() ) {
        wp_register_script(
           'ajax-quote', 
            get_template_directory_uri() . '/js/ajax-load-quote.js',
            array( 'jquery' ),
            '1.0',
            true
        );
    }
}
add_action ( 'init', 'wpse72974_load_scripts' );

Optional: The css for the loading screen

.ajax-loading-screen {
    display: none;
    visibility: hidden;
    position: absolute;
    top: 0;
    right: 0;
    bottom: 0;
    left: 0;
    width: 100%;
    background: #ffffff; /* the background of your site or the container of the quote */
    filter: alpha(opacity=0);
    -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=0)";
    opacity: 0;
    -webkit-transition:  opacity .1s;
    -moz-transition:  opacity .1s;
    -ms-transition:  opacity .1s;
    -o-transition: opacity .1s;
    transition: opacity .1s;
    z-index: 9999;
}
.ajax-loading-screen img {
    position: absolute;
    top: 50%;
    left: 50%;
    margin: -8px 0 0 -8px;
}