I've moved your post as the only commonality is exim failing and this pertains to some reason beyond changing the port. As such, it warrants a new thread.
Can you telnet on port 25 for exim?
Code:
telnet localhost 25
telnet IP# 25
In the second command, please replace IP# with the server's IP number.
Next, do you have a firewall running? If so, is it CSF or simply iptables? If you are running CSF, please disable it temporarily and recheck telnet on port 25 then. If you are only running iptables, stop the firewall temporarily and try to telnet on port 25.