As matt burns says in his answer, you may need to enable CORS on the server where the problem images are hosted.
If the server is Apache, this can be done by adding the following snippet (from here) to either your VirtualHost config or an .htaccess
file:
<IfModule mod_setenvif.c>
<IfModule mod_headers.c>
<FilesMatch "\.(cur|gif|ico|jpe?g|png|svgz?|webp)$">
SetEnvIf Origin ":" IS_CORS
Header set Access-Control-Allow-Origin "*" env=IS_CORS
</FilesMatch>
</IfModule>
</IfModule>
...if adding it to a VirtualHost, you'll probably need to reload Apache's config too (eg. sudo service apache2 reload
if Apache's running on a Linux server)