You have to provide the absolute path for your included image, as the receiver of the email does not have a file “signature.gif” in a folder “img” (that’s where the client is going to search it)
Replace
$message .= "<img src="https://wordpress.stackexchange.com/questions/266338/img/signature.gif">";
with
$message .= "<img src="http://yourwebsite.com/PATH/TO/img/signature.gif">";
Where yourwebsite.com is your website’s domain followed by the complete path.
Depending on where you added your code, you can access a WordPress function, to add the domain dynamically. This is useful if you have to move your website to another domain (for a staging/developing enviroment for example).
Have a look at get_site_url().
You could use it like this:
$message .= "<img src="" . get_site_url() . "/PATH/TO/img/signature.gif">";