I haven't used an https server before, but this might help ...
First of all, have you definitely been able to 'include' when there no images?
And then, if you add an image to the html, it then generates that error?
Is that right?
Suppose the problem is the includes ...
It says in the manual that allow_url_fopen must be enabled.
If you haven't already, try putting this page on both your servers:
<?php
phpinfo(); //(displays PHP config table)
?>
You can at least then check to see about 'allow_url_fopen' on this page. Otherwise ask the server people about it.
If it's the images causing the problem in some way ...
If it's that the images are being requested from a different server, which might be the case here (I don't know) then there is possibly a security issue.
Just like in HTML email, every time the image is requested, the server hosting the image could be activated into doing something.
This might be an issue with an HTTPS server.
Again, check with the host, if in doubt.
Hope that helps!
P.