Philip Wood gives a good introduction to the issues involved.

If instead of copper we use aluminium wire, and instead of diameter 1.0 mm we use a diameter 0.1 mm, then we get (see his answer for the formula) $I = 11$ amps which is feasible, except that this thin wire will not be able to sustain that current without heating up and melting. To calculate that we can use the resistivity $\eta$ which leads to an electrical resistance $$ R = \frac{\eta l}{\pi r^2} $$ and hence a dissipated power $$ I^2 R = \pi r^2 l \eta \left( \frac{ \rho g }{B} \right)^2. $$ Thus the dissipated power per unit volume of wire is $$ \eta \left( \frac{ \rho g }{B} \right)^2. $$ For example, with our thin wire (diameter 0.1 mm) this gives 4 watts for each 1 cm length of wire. This will very quickly melt the wire unless we can get rid of this heat, and for that you need some sort of cooling method, such as water cooling, but the water cooling will itself be heavy so the whole thing will not lift off the ground.

So now let's consider superconducting wire. Superconducting wire cannot carry any amount of current, but it can carry very much higher currents than ordinary wire without dissipating heat. This kind of wire also needs cooling, for example by liquid nitrogen or liquid helium, but once cold it only heats up owing to ordinary heat conduction, so with good insulation it will not need so much added apparatus to keep it cold. I found in a quick investigation just now that for commercial wire you can get current densities around 20 kA/cm$^2$, and a tape of cross section 9.5 mm by $1.8$ mm can carry up to 1700 amps. Putting the numbers into the formula $$ I = ({\rm area}) \times \rho g / B $$ I now get $I = 48000$ amps for this kind of wire. Unfortunately that is 28 times more current than the wire can carry. Another, thinner wire required a smaller current but again it was much more than the wire could carry without losing its superconducting properties.

However, the highest current densities that have been reported for superconducting materials are much higher, around 15 MA/cm$^2$. This suggests that in the near future superconducting tapes may become available that could carry enough current to levitate in Earth's magnetic field.

https://www.nature.com/articles/s41598-018-25499-1

Philip Wood gives a good introduction to the issues involved.

If instead of copper we use aluminium wire, and instead of diameter 1.0 mm we use a diameter 0.1 mm, then we get (see his answer for the formula) $I = 11$ amps which is feasible, except that this thin wire will not be able to sustain that current without heating up and melting. To calculate that we can use the resistivity $\eta$ which leads to an electrical resistance $$ R = \frac{\eta l}{\pi r^2} $$ and hence a dissipated power $$ I^2 R = \pi r^2 l \eta \left( \frac{ \rho g }{B} \right)^2. $$ Thus the dissipated power per unit volume of wire is $$ \eta \left( \frac{ \rho g }{B} \right)^2. $$ For example, with our thin wire (diameter 0.1 mm) this gives 4 watts for each 1 cm length of wire. This will very quickly melt the wire unless we can get rid of this heat, and for that you need some sort of cooling method, such as water cooling, but the water cooling will itself be heavy so the whole thing will not lift off the ground.

So now let's consider superconducting wire. Superconducting wire cannot carry any amount of current, but it can carry very much higher currents than ordinary wire without dissipating heat. This kind of wire also needs cooling, for example by liquid nitrogen or liquid helium, but once cold it only heats up owing to ordinary heat conduction, so with good insulation it will not need so much added apparatus to keep it cold. I found in a quick investigation just now that for commercial wire you can get current densities around 20 kA/cm$^2$, and a tape of cross section 9.5 mm by $1.8$ mm can carry up to 1700 amps. Putting the numbers into the formula $$ I = ({\rm area}) \times \rho g / B $$ I now get $I = 48000$ amps for this kind of wire. Unfortunately that is 28 times more current than the wire can carry. Another, thinner wire required a smaller current but again it was much more than the wire could carry without losing its superconducting properties.

However, the highest current densities that have been reported for superconducting materials are much higher, around 15 MA/cm$^2$. This suggests that in the near future superconducting tapes may become available that could carry enough current to levitate in Earth's magnetic field.

https://www.nature.com/articles/s41598-018-25499-1

Please note also the answer by R. W. Bird, which correctly points out that you can't make an independently flying device, like a flying carpet, this way. A horizontal coil, for example, will have a force directed up on one side and down on the other. You will have to have a wire trailing down to the ground to complete the circuit. The wire on the ground then gets pushed into the ground, while the wire in the air gets pushed upwards. Also, the device works better at the equator where the magnetic field is more horizontal. So I see this as maybe useful for moving stuff around a building site, or moving containers around a yard, or something like that, but not for transportation between cities.