You may get an error in exim_mainlog while sending an email as below,
2014-04-01 12:01:41  1Wb2Kq-0002bq-UH == firstname.lastname@example.org R=dkim_lookuphost T=dkim_remote_smtp defer (-53): retry time not reached for any host
First off, you have to ensure that remote smtp server is accepting connections on 25/tcp by telnet commnad,
# telnet host.hostname.com 25
Connected to host.hostname.com.
220-host.hostname.com ESMTP Exim 4.80.1 #2 Wed, 01 Jan 2014 11:14:46 +0530
220-We do not authorize the use of this system to transport unsolicited,
220 and/or bulk e-mail.
Next step is to tidy up exim database in case of you databases are correpted,
# /usr/sbin/exim_tidydb -t 1d /var/spool/exim retry > /dev/null
# /usr/sbin/exim_tidydb -t 1d /var/spool/exim reject > /dev/null
# /usr/sbin/exim_tidydb -t 1d /var/spool/exim wait-remote_smtp > /dev/null
Another possibility is your Email server IP Address may be blocked in Exim databases so you should unblock that.
Before that you need to take a db directory backup,
# cp -rf /var/spool/exim/db /var/spool/exim/db_bak
Stop the exim and remove ratelimit and retry and start the exim,
# /etc/init.d/exim stop
# cd /var/spool/exim/
# rm -rf ratelimit*
# rm -rf ratelimit.lockfile
# rm -rf retry
# rm -rf retry.lockfile
May remote smtp server is not accepting connections and shows below error,
# telnet domain.com 25
Connected to domain.com.
451 Temporary local problem - please try later
Connection closed by foreign host.
Need to check, may be IP address has been blocked in csf or allow the Email server IP address in both files.
# vim /etc/alwaysrelay
Once saved both files need to restart exim,
Now, you can able to send an email from server.