I need to check the routing table / firewall rules / other router rules to make sure that I'm allowed access to the copier.

Have you checked that Lanier don't provide a Mac network printing driver or tool? It would usually come bundeled with the printer.

To turn your Firewall On/Off go to System Prefrences >Security & Privacy > Firewall. You will need admin rights to make changes.

I would guess that your firewall is turned off and that this is nothing to do with your problem, but give it a go anyway!

Your Mac and the printer cannot have the same IP address. If the subnet mask is set to 255.255.255.0, then only the first three sets of numbers can be the same, while the last number has to be different for each device on the local network.

Also, the only way the Mac will "see" the printer is that if the printer supports an advertising (aka multicasting) protocol, such as Bonjour. Many office copiers don't support Bonjour so you manually add the printer by selecting the correct IP protocol (usually HP Jetdirect-Socket is a safe bet as Windows also uses the same protocol) and then entering the IP address of the printer.

You can't just guess at the IP address. If you go into your router's configuration web page, you should be able to see all connected devices. You can easily check the IP numbers for both the PC and the Mac. The odd duck out should be the printer. You can also turn off the printer, refresh the list, turn the printer back on, refresh again, and make note of the JP address that goes away and comes back. Then, assuming you have a driver or Postscript PPD file for the printer, you can. Manually enter the IP address and configure the printer using LPD.

As etresoft says you need to find out the exact IP address of the copier. Once you have you should also be able to login to the copyer setting page by typing the IP address into a web browser. It would be a good idea to change it to Static and not DHCP and pick a static IP address on the same subnet as the rest of the devices on the network but that is in the DHCP servers exclusion zone. That way the IP address will never change and you won't have to keep setting it up.

Have you checked that it is on the same Subnet as the computer you are trying to ping it from?

I.E.

If your computer has the address of 192.168.1.103 and the copier has the address of 192.168.0.98 they will not be able to talk to each other.

More than likley you router will be on the first IP address on the subnet, which will be 192.168.0.1 for example. The username and password will be something like "admin" for both or "admin" them "password" for the password. What is the router make and model, you will find it on the back of the router.

If you can't access it then you are using the wrong IP to route to it. Do you have an iPhone or iPod Touch? If so you can get an app called Scany and scan all of the IP addresses on your subnet, this will then show you your routers IP address.

If you go to http://www.default-password.info/zyxel/ it shows you all of the default usernames and passwords for Zyxel routers. Yours is not specifically listed, but i would guess it comes under the ADSL section.

Your Mac is getting a public IP address from outside of your network. Do you know why it is not getting one that is inside your router, ie 192.168.1.xyz?

Is the internet and the internal network served on two different networks? Perhaps one WiFi and one Ethernet?

Anyway, in order to connect to the printer, you must be on the same network as the printer (ie 192.168.1.xyz).

Just putting in a different octet as the last number won't connect you to the network that the printer is on. You need to figure out how to connect to that particular network and not the one you are on.

The router there might be set up to only let known computers get an IP address on that LAN. If so, you need to talk to the system admin to have your computer added to the known list.