This is happening because Exim looks at the domain, matches it to the hostname of the server and thinks that it needs to send the mail to the local server.

To prevent this "local lookup check" and just have Exim send the mail regardless, without changing the hostname of the server, you need to do the following:

1.) Open your exim.conf file (usually located at /etc/mail/exim.conf)

2.) At the end of the Routers Configuration section add the following:

myremote_smtp:
driver = manualroute
domains = !+local_domains
transport = remote_smtp
route_list = * smtp.myisp.com        # change to the desired smtp server

3.) In the Transports Configuration section, make sure it has the following lines uncommented:

remote_smtp:
driver = smtp