Aside from being a potential trip hazard, Peltier devices are very inefficient. They pump some heat from one side to the other, and introduce a relatively large heat source (from the inefficiency).
So if you don't have an effective (big, or forced convection from a fan, or liquid cooled) heat sink on the hotter side, you may end up making both sides hotter, just one a bit hotter than the other.
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.
Best Answer
The Peltier effect happens between two junctions of electrically conductive materials with different atomic lattice structures. Flow of current creates temperature difference between two junctions, but also generated the Joule heat in conductors. Also the link between two junctions is thermally conductive. So the thermolelectric element efficiency is limited by thermal "short" from cooled surface to hot surface and parasitic heat from dissipation in conductors. To increase element efficiency (aka "figure of merit", zT) the thermal conductance needs to be minimized, while electrical conductivity increased.
Unfortunately, Physics of thermal conductivity and electrical conductivity is bounded to the same internal mechanism - collective oscillations of atomic lattice, or phonons. Scholars figured out that the thermal conductivity has bottom limit for amorphous state of materials, which, unfortunately, has the lowest electrical conductivity. So the efficiency requirements for thermoelectric elements are inherently contradictory, and there is only so much one can do to improve the situation with band structure engineering using additions of different impurities.