How can the geometry of the field lines determine the type of force?

In case of an electrostatic field, the work done in moving a charge does not depend on the path taken, but only on the position of these points so the force on the charge given by Coulomb's law is an electrostatic force.

Field lines form closed loops when there is a changing magnetic field. In this case the electric field involved is non conservative. This field is different from an electrostatic field. The line integral around a closed path is not zero and when a charge moves around a closed path, the field does a nonzero amount of work on it.