THE WIRE FROM THE CONE TO THE SPEAKER INPUT KEEPS BURNING OUT

Check all the wiring connections on the speaker. Check the plug in points for the cables are not loose in the body of the speaker or the connector plugs themselves are not loose in the actual plug sockets.
Next you will need to get inside the speaker body to check the connections and soldered joints. Take a good look at the back of the cable plug sockets and to the back of the actual speaker cone connections. Check each wire at Both ends for damage. No sound at all suggests the issue might be at the 'Input' where the cable plugs into the back of the speaker body.

Tough question! WHERE are the wires ripped from? If it's directly at the speaker 'spider coil' at the speaker CONE, it's very difficult to repair but sometimes it can be done!

I have re-soldered speaker wires right at the speaker cone but FIRST I used a NEW EXACTO blade to carefully remove the enamel paint/coat from the wire end... please don't pull too hard or you will destroy the coil for good... and that's bad.

The coil shown below is toasted, but as you can see, the coil wire is quite small and delicate!

After soldering I added a DOT of epoxy fixing the new wire junction to the cone to assure that new connection/junction won't 'rattle' on the speaker as the speaker does it's job!

put a song onto play using either tape, vinyl, cd or mp3.

once playing follow the wires from the speaker that doesn't work bending it until you hear it crack or become clear if it does either of those it is the wire between the head unit and the output.

if not you can try attaching the speakers to a different head unit if it doenst work it could be the driver (the cone doesn't have to go to blow a speaker)

Audio Equipment: Is the loud popping sound when you plug/unplug your speakers bad for them?
Most speaker cones are driven from a power amplifier.. The pop - like you said - is a transient spike. But the important thing here is that it gets amplified up through the pre-amp and into the power amp.. Speakers usually move in the ol' in-out fashion with respect the the voltage at the input.. and a big spike could loosen the cone from its suspension mechanism.. The higher the power and the more delicate the cone - the more the danger to the speakers ( what is normal for a woofer cone, could end blowing up a tweeter cone) . For e.g. In live events, where high output PA speakers are used, this is a very serious issue - because such a spike could throw the speaker cone out of its place, and it wouldn't work from that point on. For this reason, some high end audio equipment have a limiter at the output, that prevents signal from exceeding a certain predefined limit. So any occuring transients would never hit the cone at a level that is unhealthy to them.

Most home audio equipment tends to operate at lower levels. That being said, there is still some damage inflicted on the speakers - just maybe not of the same magnitude.. especially the smaller ones.

If your system has a remote control, it might be wise to hit a mute button just before you change to another input... or maybe find a way to turn off just the speakers (if separately powered) before you interchange the connection

If I understand you, you're observing the speaker cone moving in one direction without any input. That would say the internal amp is malfunctioning and pushing DC to the speaker. Beyond that I can't say, but it is endangering the driver.

use a AA battery and touch speaker wire 1wire on positive other on negative
look at speaker if cone come out or up wire to positive end is positive
if speaker cone goes in or down the wire on the negative of battery is positive

All you need is speaker wires or lamp electrical wires. Go to your local hardware store and ask for 16/2 AWG wire

there isnt really much you can do, you can try and solder it but you take a chance of burning the speaker. it is best just to replace!

It may be the wire from the amp to the speaker hitting the cone.

You will have to pull the speaker to check this.

sounds like you are frying the voice coil! Push the centre of the cone in and if you hear a crackle then time to buy new speakers.. sounds like they are on way out if you can smell burning!