Would a compass with unmagnetized needle work? We know that the needle that is used in a compass is a permanently magnetized ferromagnetic material and commonly steel is used.
If we used an unmagnetized iron needle instead, would it still align with Earth's magnetic field lines? If yes, how?
 A: A magnetic dipole would be induced in the iron bar and the iron bar would try and align itself along the magnetic field lines because of the torque applied on it by the interaction of the induced dipole and the Earth’s magnetic field.
However since the torque which was applied on the iron bar would be very small the chances are that there would not be an alignment even if you waited a long time.
A: Probably yes, in a careful experiment. 
A paramagnetic material like aluminum will align with a strong magnetic field and diamagnetic metal plates align themselves perpendicular to the field.
Magnetically weak iron has a relative permeability many orders of magnitude larger than paramagnetic or diamagnetic materials. This probably offsets the much weaker field. A needle or plate would likely align itself parallel to the field.
Shape anisotropy ensures that the magnetization is mostly parallel with long axes.
One experimental problem may be to rule out the effect of possible areas with remanence. So a careful degaussing would be necessary.
A: For an unmagnetized iron needle to align with an external magnetic field, the field would need to be able to induce a magnetization in the needle. This is definitely possible with a large enough field.
If a naturally ferromagnetic material is unmagnetized, it still contains small magnetic domains inside. However, the sum of the magnetizations of all the domains is zero. If you apply a strong enough external field, the domains will align to the field. The following image is from the Wikipedia page on magnetic domains (https://en.wikipedia.org/wiki/Magnetic_domain).

Then the question is whether the Earth's magnetic field is strong enough to realign the domains in an iron needle. The Landau Free Energy is used to determine this, as the domains will align in whatever way minimizes this energy. Parameters that determine this energy include things like: size and shape of the needle, material (in this case iron), and external field strength.
If the external field is strong enough to cause magnetization, the direction of the induced magnetization will be in a direction that minimizes the anisotropy energy and is pre-determined by the dimensions of the needle. The dimensions give rise to an "easy" axis, meaning the free energy is lowest when the magnetization is in a particular direction. In general, this axis could be in-plane in the x or y direction, or perpendicular to the needle in the z direction. In a graph of energy versus angle of the magnetization from the easy axis, there will be two energy minima: one along the easy axis, and another at 180 degrees (still along the easy axis, just pointed in the opposite direction).
Anyway, I haven't done the calculation, but I don't think the Earth's field is strong enough to cause realignment of the domains. I would also like to mention that once the needle has been magnetized, if you remove the external field, the 
 needle will keep its magnetization. It would take the addition of a lot of energy to reorient the domains/magnetization that could come from a new external field, or even thermal energy.
Edit: If your question is more about the torque that the needle would experience, it would follow the following equation assuming it was indeed magnetized:
$\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B}$.
$\bf{m}$ is the magnetic moment and is related to magnetization, $\bf{M}$, by: 
$$\mathbf{m}=\iiint\mathbf{M}\ \mathrm{d}V$$
For more information, here are some resources:


*

*https://en.wikipedia.org/wiki/Magnetic_domain

*Magnetism and Magnetic Materials by J.M.D. Coey. Sections on Landau Free Energy, magnetic moment, and maybe even the Stoner-Wohlfarth model would be enlightening.

*https://en.wikipedia.org/wiki/Magnetic_moment
A: There is a way of making a compass by floating the needle on a leaf in a water.
Weather it works is to be verified in practice. In a 'hard' water with small enough needle this could be achieved even without the leaf.
