Is the Lorentz force conservative? Is the Lorentz Force acting on a wire, that has current $I$ in a magnetic field $B$ conservative? Or non-conservative? I understand that all the fundamental forces are conservative, am I correct? 
 A: The Lorentz force 
$$
\vec F = \frac{d}{dt}\vec p_\text{charge} = q ( \vec E + \vec v \times \vec B )
$$
is in general not conservative, because it is possible to construct a system of two charges where the forces are not equal and opposite.  Conservation of energy and momentum in electrodynamics requires you to also consider the energy and momenta associated with the fields.
Keep in mind that the Lorentz force on a moving charge $\vec F = q\vec v \times \vec B$, never does work. If magnetic field causes a current-carrying wire to accelerate, the energy comes from either the power supply controlling the magnetic field $\vec B$ or the power supply maintaining the current $\vec I$, and the work done must be computed using the induced electric field $\vec \nabla \times \vec E = -\partial \vec B / \partial t$.  This is a very different creature from, say, a mass oscillating on a lossless spring, where the kinetic energy of the mass and the potential energy of the spring convert into one another.
A: 
Is the Lorentz Force acting on a wire, that has current I in a magnetic field B conservative? Or non-conservative?

As the wiki page you linked to explains, conservative force is a force that acts on particle and work done by this force when the particle moves from A to B does not depend on the path, just on the points A,B.
Wire is not particle, so the original meaning does not apply. If you want to use the word conservative for wire, you'll need to define what it means.

I understand that all the fundamental forces are conservative, am I correct?

Electrostatic and Newtonian gravitational forces are conservative. If the particles in the source of the force move, electric force will not be electrostatic and  thus won't be conservative.
