This system is defunct and no longer in use by my building.
Id like to get rid of the hideous phone on the wall.
Can I cut these? Is there an order in which to cut them?
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Phone wiring is mostly low voltage, though old-style electromagnetic ringers can briefly kick it up to 90v when ringing. If you really want to just discard this, I'd just cut off the bared copper ends, fold the wire back on itself once so the ends are against the outer sheathing, give it a quick wrap with electrical tape to hold them there (just once or twice around the ends), and shove it back into the wall.
That may not be the best answer. But it would work, and be fairly easy to undo if you discover you need them after all.
If it were me I would just unscrew the plate and pull it to see how much extra wire is in the wall. Cut all the wires close to the connectors with insulated wire snips - if there is enough extra wire inside the wall to reconnect it, you could snip the (most likely) UTP cable again so there are no dangling wires. If there isn't, wrap the small wires around something round like a screw driver handle to shape them neatly, remove the handle and put electrical tape around the bundle.
As others recommended, a voltage tester is probably a safe precaution to make certain there isn't any live power running through those wires.