Classically, electrons collide with other electrons and massive (by comparison), stationary positive ions as they conduct down a wire when an electric field is applied. Is there a good mechanism according to classical theory for the collisions between the electrons and positive ions? Is said mechanism electron scattering? If so, why would these oppositely charged particles only ever approach one another but never actually touch (which I assume is how they behave)?
The collisions between the electrons and the ions in the wire is not electron scattering. Electron scattering is when the electrons bounce off from a crystal lattice of atoms, displaying wave characteristics. That is a completely different thing right there.
First of all, even within a lattice, the electron has a lot more free space to go undisturbed. But since we are talking about electrons from every one of those atoms, then surely some time or the other, the electron would get close to some nucleus and lose a bit of their kinetic energy. So, the electron's speed goes down and the direction is changes due to the attraction from the nucleus. I suppose in many situations, there are as many electrons that travel through the lattice undisturbed as there are those electrons that are disturbed in their trajectories and hence slowed down - or even absorbed.
The heat is due to the conservation of energy as the nucleus literally takes that lost kinetic energy.