I am installing a new ethernet switch, and the instructions say to connect the ground screw to the rack or some other kind of electrical ground. What are the risks of not ensuring that equipment is properly grounded? Is it because of concerns for the equipment's lifetime, my lifetime, or both?
This is because that equipment was designed to be installed anywhere in the world, and not every place has 3-wire electrical. Additionally, not every 3-wire install has a well grounded ground-wire, so having the option of creating a separate ground is a very good idea for the enterprising electronics manufacturer.
Since you have a good ground with your power, it won't matter.
Potentially significant. If both grounds have different ground potentials, you can cause current through your device. This is potentially harmful.
If the switch is not connected to the ground, a power surge within one of your cable runs (whatever induced it) is likely to spread across your entire cabling plant, surely inducing transmission errors and possibly reducing all connected equipment's lifetime as well as the lifetime of the users currently touching the cable pins. See this anecdotal blog post about what could go wrong.
However it is uncommon for a rack-mountable switch to come with a 2-pin power connector where electrical safety would be a major concern. At least the European models are using the 3-pin IEC 320 C14 plug where one pin carries electrical ground. Connecting the switch to the common rack cabinet / cable plant ground (when using shielded cabling) will help reducing stray electric currents on the ground which otherwise may affect signal quality.
Well, consider you do not ground it properly, you add an ethernet cable and there is some leakage. Chances are you will get electrocuted and indeed you might need temporary or permanent replacement.
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So in short, not grounding could shorten you and your equipment's lifetime.
Let me tell you my story of why grounding is important. In my office there was an un-grounded outlet, I did not know this at the time. I was troubleshooting a PC that would only partially boot. In the process my I was reaching behind the PC to plug in a USB keyboard and my arm ended up touching the case. I ended up with a nasty shock and a renewed respect for electricty.
Now with that story out of the way as long as your outlet is grounded correctly there is little need to use the ground screw. However if you are not running from the wall outlet (using DC power perhaps) the ground screw is an important piece to the safety of you and your equipment.
If you have a shielded cable connecting two pieces of equipment that are not properly grounded to a common ground, the shield will complete the circuit between the two grounds. The amount of current that can flow can be surprisingly high. There's a well-known, well-documented story of an RS232 cable run between two buildings that started a fire.
With standard Ethernet cabling, it does not matter. The Ethernet signals are all isolated from the ground (typically by a transformer). This was an intentional design decision and one of the driving factors was to allow Ethernet to tolerate a lack of a common ground.
However, any cable that includes a ground wire that is not a power cable should only connect two devices with a common ground connection.