Does current have an effect on the wire it moves through? I think I heard somewhere that over time small wires in CPUs can move because of the current going through them. If it is a thing what's the technical term for it?
Yes, at the nano-meter scale you always run into an issue titled Electromigration.
At the nano scale a conductor has to not only be rated for voltage drop but also for current density. To achieve sufficient reliability due to deterioration from electromigration.
The vast majority of the time, the physical effect of a current through a wire is negligible.
High current densities can physically effect a wire in 4 very different ways:
The electromigration effect ( as mentioned by sstobbe ) is the gradual movement of ions along the length of a conductor due to the momentum transfer of electrons moving in a wire. At high current density, eventually enough ions of the metal wire get pushed out of where they were originally placed to leave a gap (an open circuit) in that wire, or build up enough displaced ions to bridge the gap between that wire and some adjacent wire (a short circuit). This can cause long-term reliability issues -- when people design things to run right on the edge or slightly past manufacturer's recommendations, this effect allows the device to seem to work fine for a while, but lead to early failure.
The resistive heating of wires, also known as Joule heating or "I squared R heating", is proportional to the square of the current through the wire. When an integrated circuit is stressed by voltages far beyond the manufacturer's recommendations, this effect often melts or vaporizes one or all of the small wires of the IC.
The magnetostriction effect causes some materials to change shape when magnetized. This effect is the most common cause of transformer hum. (The current through the coils of the transformer creates a strong magnetic field, which leads to magnetostriction in those wires or in the ferrite core or both).
In a strong magnetic field, the wire is physically shoved sideways by a force proportional to the current and to the magnetic field -- that force is sometimes called the Laplace force, but more often calculated as part of the Lorentz force (as alluded to by Uwe ). This effect also causes transformer hum.
If strong currents flow in two parallel conductors, the magnetic fields cause a force between the conductors, see https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_force_law. But the currents flowing through small wires in a CPU are too small for movement.