Can’t connect to a Datamax-O'Neil E-4205A E-Class thermal printer when it is hooked up via Ethernet via a Netgear GS108E managed switch

I've just recently bought a Datamax-O'Neil E-4205A E-Class thermal printer off of eBay, and I'm having some issues trying to connect to my network. Using direct Ethernet, it connects fine and is discoverable if I put the printer's IP into a browser. But I need to use a network switch to make the distance over to where it is going to be living. Using a Netgear GS108E managed switch, I can't get it to connect at all to the network. According to the switch, it's sending and receiving packets, but I'm getting nothing when I put the IP into a browser or when I print a test page off of the printer.

Question by Riley on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.