Defer the loading of images to the production server. Use either Apache or NGINX depending on your local server setup.

example-nginx.conf

################################################################
# Example configuration file for nginx
#
# To add a new local WordPress domain to your environment, copy
# this file using a filename that matches the domain you wish to
# setup. For example - mylocaldomain.com.conf would be an ideal
# filename for http://mylocaldomain.com
#
# Once copied, you will need to modify two settings in the server
# configuration provided:
#
# 1. server_name - Change the server_name parameter in the server
#                  configuration below to mylocaldomain.com
# 2. root        - Change root to the full path that your WordPress
#                  site lives at inside Vagrant. An example would be
#                  /srv/www/mylocal-wordpress
#
# You do not need to worry about modifying the listen or include
# parameters as those are the same across all test sites for most
# basic WordPress configurations.
#
# Once your new domain has been added, make sure to restart the
# nginx process by running `vagrant provision` in your local
# environment or `sudo service nginx restart` after `vagrant ssh`
################################################################
server {
    # Determines the port number that nginx will listen to for this
    # server configuration. 80 is the default http port.
    listen       80;

    # Listen for HTTPS requests as well
    listen       443 ssl;

    # Tells nginx what domain name should trigger this configuration. If
    # you would like multiple domains or subdomains, they can be space
    # delimited here. See http://nginx.org/en/docs/http/server_names.html
    server_name  gregrickaby.dev;

    # Tells nginx which directory the files for this domain are located
    root         /srv/www/gregrickaby/htdocs;

    location ~ ^/wp-content/uploads/(.*) {
            rewrite ^/wp-content/uploads/(.*)$ http://gregrickaby.com/wp-content/uploads/$1 permanent;
    }

    # Includes a basic WordPress configuration to help with the common
    # rules needed by a web server to deal with WordPress properly.
    include /etc/nginx/nginx-wp-common.conf;
}

gistfile1.txt

# tl;dr, here are the two rewrite rules needed depending on your server setup

# Apache .htaccess
RedirectMatch 301 ^/wp-content/uploads/(.*) http://livewebsite.com/wp-content/uploads/$1

# Nginx
location ~ ^/wp-content/uploads/(.*) {
  rewrite ^/wp-content/uploads/(.*)$ http://livewebsite.com/wp-content/uploads/$1 redirect;
}

if you need to include a url with http authentication just use user:password before domain, eg:

# Apache .htaccess
RedirectMatch 301 ^/wp-content/uploads/(.*) http://user:password@livewebsite.com/wp-content/uploads/$1

# Nginx
location ~ ^/wp-content/uploads/(.*) {
  rewrite ^/wp-content/uploads/(.*)$ http://user:password@livewebsite.com/wp-content/uploads/$1 redirect;
}