How is the curl of the electric field of a dipole zero? For a static charge, the curl of the electric field is zero. But in the case of a static dipole the electric lines of force curl. How it that possible? 
 A: The electric field of a dipole has zero curl; this is easy to verify because it is (the $d\to0$ limit of) a superposition of two monopole Coulomb fields with zero curl. If you want something more explicit, then simply start with the explicit electric field,
$$
\mathbf E=\frac{1}{4\pi\varepsilon_0} \frac{3(\mathbf p\cdot\mathbf r)\mathbf r-r^2\,\mathbf p}{r^5}
$$
and calculate the curl $\nabla\times \mathbf E$; you will find that it's zero.
You do provide an interesting observation, though, in that

in a dipole the electric lines of force form a closed path from a positive charge to negative charge,

and this is indeed true: if you start your curve just above a point dipole, and loop around to just below it, then that finite segment will accumulate a nonzero line integral. However, to have a closed loop, you will need to cross directly across the dipole itself, and this will introduce a singularity into the circulation integral. This essentially breaks the game and none of the calculus applies any more. (Similarly, you can't cheat and go around the dipole, either, because the field will be very strong and point against the line element, so the circulation integral will be exactly zero.
A: I think I do not understand the equation. If you try to find the electric field for a static dipole you have two main way: starting from a potential and then you make the approximation of great distance so that the multipole expansion is truncated to the second order in the charges, or you can compute the electric field generated by two charges very close one to each other. In the first case, in order to find the electric field you will take the gradient of the potential and then you are already imposing that the curl of the electric field is zero and all the charge are stationary. In the second case you are calculating the electric field very far away from the charges and then the electric field is the superposition of the electric field of each charge. The electric field of a charge has null curl, so, since the curl is linear, the electric field of the dipole is zero. 
If you go very close to the charge, in order to find the electric field I think you have to consider higher order in the expansion of the multipole, which decay faster when you go do great distance. However if you suppose that the charge are still and they do not attract or repel to each other, no current can be generated and so, there is no variation of the magnetic field. From the Maxwell law you always get zero of the curl of the electric field in vacuum.
The dipole approximation is, in fact, the hypothesis of two charge, far away from each other in such a way that they do not collapse or go away from each other, but that very far away you can consider form a single entity. No current are involved, so no dynamic magnetic field, so no curl of the electric field.
