If you are getting an error with the following error
SMTP error from remote mail server after RCPT TO:<admin @domain.com>: 550 smtp auth requried
from the script that is running on your server but the domain.com isn't locate on the same physical machine. You are most likely doing a SMTP Relay and your exim isn't really happy with not having authentication credential being provided.
In this case, you can just add the following to your exim.conf file, assuming 126.96.36.199 is your script server,
domainlist local_domains = dsearch;/etc/exim4/domains/ domainlist relay_to_domains = dsearch;/etc/exim4/domains/ hostlist relay_from_hosts = 127.0.0.1 : 188.8.131.52 hostlist whitelist = net-iplsearch;/etc/exim4/white-blocks.conf hostlist spammers = net-iplsearch;/etc/exim4/spam-blocks.conf
you can also placed 184.108.40.206 into the file /etc/exim4/white-blocks.conf to whitelist the host on your server.
you might also need to add auth_advertise_hosts = * as show below,
host_lookup = * auth_advertise_hosts = 220.127.116.11 rfc1413_hosts = *
which expand to all host other than localhost (of course, you might want to change it to ip instead of *)
this should allows your script to sent email using your smtp server as a relay without the need for authentication.