A device like a Peltier cooler which transfers heat to another surface not in direct contact with the cold side?
This isn't going to work. 

Metal is an excellent conductor of heat, so its temperature quickly becomes the same throughout. The figure of merit for a thermoelectric device is crippled by high heat conductivity [\$\kappa\$].

$$ZT = {S^2GT \over \kappa}$$

Additionally, copper has a Seebeck coefficient [P] of 1.5 μV/K. The most common material used, bismuth telluride, has a coefficient of -297 μV/K. Notice this is the squared term in the numerator, it's very relevant. This is why peltier devices are not made of metal. Increasing the current will only waste more energy.

Result, you get a heater.



Ok, you say, I want my wire to be made of the same material inside a typical junction. 

The problem there is the thermoelectric effect has a limited efficient distance for any material. As the thickness of the junction material increases the electron mean free path doesn't. Instead of heat transfer you get heat generation (see Fourier's Law).

Result, you make a heater.



Ok, you say, I will make many small stacks of peltier junctions in my wire passing heat along like a bucket brigade. 

The problem here is each device generates its own heat while trying to pass along heat from cold to hot sides. Stacking devices quickly becomes very inefficient and things get hot very quickly.

Result, you make a heater.