Are electric field lines due to Faradays law closed? Are the field lines of electric field produced by Faraday's law of induction  (and assuming $\rho = 0$) necessarily closed? If not what would be a counterexample. If it's true, how to prove it in  mathematical rigorous way?
How does the situation changes, if you consider the net field produced by a charge distribution $\rho$ and by Faraday's law of induction (where the latter one must not be zero)? Again, what would be a counterexample or how to prove it?
Do you have any references for this?
Edit
How does the situation changes if you consider only cases where the induced electirc field is time independent? Is the claim true for this case or is there also a counterexample as given by @arccosh?
 A: Counterexample: An electromagnetic plane wave in free space (no charge or current) at a fixed frequency and linear polarization has oscillating electric field in one (unsigned) direction normal to direction of propagation. These field lines are not closed.
Note: Although Maxwell's equation is "Faraday's equation plus other stuff", adding more equation actually constraints to the problem more, which is okay if you're just seeking a counterexample.
Addendum for comment 1: From $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$, we see that curl of the electric field is nonzero, since magnetic field is changing. For concreteness, say that we have
$$\vec{E} = \vec{E}(z) = \cos(kz - \omega t) \hat{x},$$
then at $t = 0$ the rectangular path
$$(-L,0,0) \to (L,0,0) \to (L,0,\pi/k) \to (-L,0,\pi/k) \to (-L,0,0)$$
for large $L$ would have non-zero path integral for $\vec{E}$.
Introductory texts uses "closed field line" to motivate conservative vector field, where path independence of line integrals allows scalar potential to be defined. 
For the non-conservative field, closed field lines (say, electric) provides a nice intuition: if a wire loop is placed along the closed field lines, then we would obtain current and power can be extracted. However, this can be done even if the fields lines are not closed, since the wire form a "fixed track" for current to travel. In line integrals, this is described by the use of dot product.
As for conditions to obtain periodic orbits (for every point) in a general 3D vector field, I don't know the answer, but that's a dynamical systems question that's more appropriate for a math forum.
For the 2D case, loops can arise from contours of equipotential lines of a potential. It turns out that this corresponds to the divergent-free condition.
