Sheetrock was put up on both sides of a wall where our thermostat wires were located.
Is there anyway that we can locate the end of this wire without cutting into the sheetrock at various places? The wire is curled up in the wall where the thermostat was to be located.

Can you localize it to one wall cavity by seeing where it goes up through the floor? If so, I would cut a hole about 4' up the wall just small enough to hide with the thermostat you intend to install. Fish around inside with a coat hanger wire or something, and maybe you will be lucky enough to snag it. If not, try a mirror and a flashlight.

You could also try top locate it with a stud detector or with the power on, a voltage detector*. Once you know where to fish, it may work better. Either may help if you don't know which wall cavity.

*To do simple checks like this you do need some tools. A test light, a meter, or a voltage detector might be the best place to start with. I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit. You do not have to open up housings and expose electrical contacts. You are looking at where your hand is, not where the meter is. Most people are capable of doing repairs and will get it going and not get hurt if they use a little sense. The voltage detector makes it even easier.

Apply 24 volts to any one of the wires at the furnace and use a tick tracer, about 30 dollars at Lowe's. But it will show you all live wires, 120 and low voltage.

Can you localize it to one wall cavity by seeing where it goes up through the floor? If so, I would cut a hole about 4' up the wall just small enough to hide with the thermostat you intend to install. Fish around inside with a coat hanger wire or something, and maybe you will be lucky enough to snag it. If not, try a mirror and a flashlight.

You could also try top locate it with a stud detector or with the power on, a voltage detector*. Once you know where to fish, it may work better. Either may help if you don't know which wall cavity.

*To do simple checks like this you do need some tools. A test light, a meter, or a voltage detector might be the best place to start with. I came across the niftiest gadget for trouble shooting, a voltage detector. They work through the insulation of wires. There are several brands. I have a GB Instruments GVD-505A, less than $15 at Home Depot. Touch it to a hot wire, and the end glows red. Find the doodad that lights it on one side, and not the other, and you have the culprit. You do not have to open up housings and expose electrical contacts. You are looking at where your hand is, not where the meter is. Most people are capable of doing repairs and will get it going and not get hurt if they use a little sense. The voltage detector makes it even easier.

Yes, I also posted to this, I like the better tracer with a sensitivity adjustment. Makes low voltage wires easier to find.