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 [16059] courier_login authenticator failed for (domain.com) []:47845 I=[]:25: 535 Incorrect authentication data (set_id=support@newdomain.com)

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.

Solution 1:

Open the exim.conf.local file and update,
vim /etc/exim.conf.local



Solution 2:

Login to WHM panel , Main >> Server Configuration >> Tweak Settings >>
Restrict outgoing SMTP to root, exim, and mailman (FKA SMTP Tweak) :

Make above function Set to : OFF

I hope you have fixed this error permanently.