You may get an error in exim_mainlog while sending an email as below,

2014-04-01 12:01:41 [10040] 1Wb2Kq-0002bq-UH == 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 25
Connected to 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

#/etc/init.d/exim start

May remote smtp server is not accepting connections and shows below error,
# telnet 25
Connected to
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

# /etc/senderverifybypasshosts

Once saved both files need to restart exim,
#/etc/init.d/exim restart
Now, you can able to send an email from server.