The claim, "the magnetic force does no work," while technically true, is misleading in so many situations and actually helpful in so few that we probably shouldn't put as much emphasis on it as many textbooks do. The magnetic field stores energy, and a change of the configuration of a system (like moving a current) can convert that stored energy into other forms, so the system will certainly behave as if the magnetic field is doing work, but careful analysis will always show that it's not actually the magnetic force that's doing the work associated with the change of magnetic energy. Usually it's an electric force.
In your example, I think the subtlety lies in assuming that $I\vec{\ell}$ for the current $ = q \vec{v}$ for individual charges. The charges will tend to change trajectory, i.e. $\vec{v}$ changes, due to the magnetic field, but $\vec{\ell}$ maintains its direction. Why? Because the wire (magically? We never really explain it very well in class) constrains the charges to move along its length in spite of their lateral movement produced by the magnetic field. As others have said above, the wire exerts a force on the charges to keep them in a straight line, and so the charges exert a reaction on the wire which moves it.
So technically, the magnetic force isn't doing any work on the current; it's just redirecting the force that would otherwise be pushing the current down the wire so that that force has a component pointing perpendicular to the wire instead.
Now for the bonus question: a magnetic dipole, e.g. a current loop or a small refrigerator magnet, will experience a force in a nonuniform magnetic field (though not in a uniform one). The dipole will then accelerate if this is the only force acting on it, so work is being done on it. If it's not the magnetic force doing the work, then what is it? (Really, as in this case, it gets silly sometimes to insist the magnetic force isn't doing work. For many intents and purposes, you might as well treat it as though it is.)
Best Answer
The resolution to this problem is simple once you know how...
Remember work done is force times distance moved in the direction of the force. The electrons are moving upwards, the Lorentz force $-ev \times B$ is in the direction shown in the diagram. BUT, the force did not do any work, because the force is perpendicular to the direction of travel of the electron!
What has happened is that the electron has been merely deflected sideways (but its energy has not changed). This deflection means there is now a slight imbalance in the density of charge because the electrons have all been slightly shifted to one side. That means there is an electric field induced (via Gauss Law)...and it is this electric field, which is also in the direction shown by $F$ in the diagram, that ends up doing the work on the electrons.