If you unable to send an email from Exim server, you need to check in Exim main error log, it’s generated below errors.
tail -f /var/log/exim_mainlog
2014-07-08 14:36:38  courier_login authenticator failed for (domain.com) [192.168.0.21]:47845 I=[192.168.0.20]:25: 535 Incorrect authentication data (firstname.lastname@example.org)
It occurs may updated your cPanel in your server, cause there are few new functions updated for Exim Mail. They have added the function which rejected mail authentication.
Open the exim.conf.local file and update,
Login to WHM panel , Main >> Server Configuration >> Tweak Settings >>
Make above function Set to : OFF
I hope you have fixed this error permanently.