Can current though a long wire increase its mass?

Current flowing through a wire does increase its mass, but the increase is minuscule, and it's not due to the increase in the speed of the conduction electrons.

Those electrons behave in ways that are similar to the molecules in a gas, and we can successfully model many properties of metals using the Fermi gas model.

In a wire with no electrical current flowing through it the conduction electrons are bouncing around randomly with the Fermi velocity (around $1.57×10^6$ m/s for copper and $2.03×10^6$ m/s for aluminium) but their mean velocity is zero. This is just like the molecules in still air: each molecule bounces off its neighbouring molecules at high speed (around 300 to 400 m/s for oxygen & nitrogen at room temperature) but because the overall motion is random the whole parcel of air has zero total velocity.

When current is applied to the wire, the electrons acquire a small amount of extra velocity, but that drift velocity is very small. Wikipedia gives a value of $2.3×10^{-5}$ m/s for a 2 mm diameter copper wire carrying 1 ampere. Clearly, such a velocity is far too small to produce any relativistic effects.

However, a wire that's conducting electricity is carrying electromagnetic energy, and that energy contributes to the (rest) mass of the wire, but it's still very tiny. In what follows, I'll use the approximation $c=3×10^8$ m/s.

Let's say we have a span of cable 300 m long carrying 1 megawatt, that is 1 million joules / second. The energy travels through the cable at close to the speed of light, let's be generous and say that it's travelling at $c$. So it takes 1 microsecond for electromagnetic energy to travel from one end of the cable to the other, so the total amount of energy "in" the cable at any given moment is 1 joule. (The energy isn't exactly inside the cable, it's in the electromagnetic field surrounding the cable, but it's still essentially being carried by the cable). Thus the mass of the cable is increased by the mass equivalent of 1 joule, which is $1 ,mathrm{joule}/c^2=1.11×10^{-17}$ kg or 11.1 femtograms.