For Sun (solaris)
Changing IP address:
You will have a file called /etc/hostname.INT where INT is the interface ID that your network cable connects to. It is probably the only hostname.INT file you have, if you only have one network connection. In that file you have an IP address or a name (default is name, but if it is IP you can put the new IP here). If it is the name that name will be in your /etc/hosts file. Edit your hosts file to point the name to the new IP address. If the IP address will be on a different subnet you will need to edit /etc/netmasks and set it there. Every time you reboot thereafter the new config will take effect. If you want to activate it without rebooting you can do this - ifconfig INT inet NEW_IP netmask NEW_NETMASK broadcast NEW_BROADCAST up. If the machine will be on a new subnet you may need to set the default route. You can do this permanently (for reboots) by putting the new default route in /etc/defaultrouter. To set it on the command lineto take effect immediately use - route add default NEW_ROUTE. Don't forget to delete the old default first. (You can see the old default with - netstat -rn |grep default). To delete it use - route delete default DEFAULT_IP.
As for DNS, otheranswers are correct. You won't need to change the MX record if only the IP address is changing, but you will need to change the A and PTR records that associate the server name with its IP address.
On HP-UX you set your permanent (for reboot) network settings in the /etc/rc.config.d/netconf file as will as making changes to the /etc/hosts and /etc/netmasks file where needed. SAM can step you through these procedures if you prefer this over editing the files. In AIX you can use SMIT. Don't forget the default route. If you have another flavor of UX let me know and I'll post the proper method.
Good luck.