It looks to me like the switch on the right is a former 3-way. The black and red labeled "no voltage" were the messengers, and the black labeled "voltage" was (is) the supply.
But we're not sure about the messengers or what they connect to. If you can find the other end of the messengers, and establish that they are messengers, I would wrap yellow tape around all 4 wires (2 wires x 2 ends) to mark them as messengers. They sell 5-packs of colored electrical tape for a few bucks. Don't mark them unless you can mark both ends, or you'll just make yourself confused later.
It looks like they did the trick of baring a part of the wire 6" before the end of the wire, connecting it to a screw on switch 1 (right) and then continuing it to switch 2 (left). That will be a little hard to put on a wire-nut unless you cut it in the middle of the stripped area. Which would be fine.
The other non-ground wires need to be capped, period. You have no idea what they go to, and when you crunch everything back into the box, you don't want them touching a hot and "lighting something up" in the bad way.
Are you sure a switch out there somewhere hasn't lost power? There must surely be a partner 3-way to this one. If you now find a switch in your house doesn't work, take it apart and see if it is the partner 3-way to this one. If so, mark the messengers (disregard position, messengers are the brass screws), then at this end, connect one messenger to supply and leave the other messenger capped. At that point the other switch will operate in one position or the other. If it lights in the "down" position and you find that annoying, come back here and switch messengers.
Lastly I see the screws are all the way out on your remaining switch. That, and the hack-scratches on the loose wires, makes it look like the installer used the "back-stab" wiring method. That is a cheap builder trick to make wiring faster, but experience shows it is unreliable in the long term. I recommend converting to either side-screws or go out and get the nicer Leviton "screw-and-clamp" type receptacles.
You have a lost neutral. Stop using the circuit until fixed.
This is a classic "lost neutral" we get all the time either on main panels or multi-wire branch circuits (MWBC, which is what this is). The whole thing is one large circuit since they share neutral. MWBCs are complicated and have special rules that need to be honored if you don't want things like this happening. Given the casual nature of speech and this question blowing over from ee.se, I suspect you may be out of your depth, no offense.
Particularly if "melting wires together" is a thing that is happening to you, it's really time to put the tools down and call a pro, or else "get religion" about doing good electrical work to Code and then read a book on electrical cover to cover and not skip through it as people with "smartphone attention spans" are prone to. Google cannot substitute for a book because it only answers questions,and your field knowledge is not sufficient to know which questions to ask.
This is a 3-phase MWBC, implying New York City residence. My advice on MWBCs here applies. Particularly the part about pigtailing neutrals. You are 3-phase so where it says 2, read 3.
The "symptom" for this problem is that if you plug a bulky load into one leg of the MWBC, say a resistive heater, and measure voltage at other legs of the circuit, it will be other than the expected 120V. Another giveaway is a high difference in voltage between neutral and ground, on a 100' run, 1-2 volts is to be expected but no more than that.
It sounds like you did not fix (or find) the problem with the neutral. If you insist on continuing to work on it, and you follow my advice, you'll be revisiting a lot of it, but also look in the service panel at the neutral termination there. The overcurrent may have fried it.
Best Answer
If the "voltage tester" is one of those "tick tracer" type that don't actually need to touch two wires (or one wire and ground), they function my simply detecting magnetic fields AROUND wires. So they will "light up" by just being NEAR wires, even wires BEHIND something else. Basically, it doesn't mean anything in the scenario you described.