Exim here is your sendmail facility. Normally you use it to send email. Not receive email. You probably don't receive email with Exim, as you sensibly use a Dreamhost IMAP account instead.
This is a typical configuration for a Debian system.
View your log:
tail -f /var/log/exim4/mainlog
Create a mail user http://panel.dreamhost.com/
Your smtp relay will probably be mail.example.com. In this example, mine is mail.dabase.com.
From Debian Etch configuration (site down, please provide mirror) setup a satellite configuration and edit:
And put in your Dreamhost mail username and password.
Test like so on a shell:
echo Please work | mail -s "testing" firstname.lastname@example.org
Note it does take a bit of time for Dreamhost to set the mail account, so your mail relay might be not authorising at first.
You might need to restart the service too:
sam$ sudo /etc/init.d/exim4 restart
Setting up 587
Optionally, you probably want to append port = 587 to the split configuration file:
As many ISPs block the normal port 25 as they think you're trying to relay spam. More info: http://wiki.dreamhost.com/index.php/KB_/_Email_/_POP3/IMAP/SMTP_Servers#Does_my_ISP_block_port_25.3F