Can electric field lines form close loops in EM?

A closed loop of $bf E$ never happens in static problems. But in dynamic problems you can get closed loops of $bf E$. The standard case is a region of space where there is a steadily increasing magnetic field. The electric field in this situation is in loops around the region where $bf B$ is increasing, in a similar pattern to the one you see for $bf B$ around a current-carrying wire.

To prove the above one can use Maxwell's equations, which are differential equations for these vector fields, but I am assuming you don't want that level of mathematical detail.