If you look at elementary examples, it seems like a Lorentz transformation takes a field pattern with a lot of straight field lines to another field pattern with a lot of straight lines. Examples:
an electromagnetic plane wave, boosted longitudinally
a parallel-plate capacitor, boosted along any axis of symmetry
the field of a point charge that moves inertially
At first when I thought about this, I convinced myself that certainly it must be true that Lorentz transformations take straight field lines to straight field lines, and I just needed to find a proof. As I thought about it some more, I started to doubt whether it was true, and then whether it was even a well-defined statement. In general, if I show you the two field patterns, it isn't even defined which field lines map to which ones. I suppose the right way to state the conjecture might be the following:
Conjecture: Given an event P, suppose that the electric (magnetic) field line through P is straight (for some finite length), and the magnetic (electric) field is zero in some neighborhood of P. Then under a Lorentz transformation, the electric (magnetic) field line through P is again straight.
Is this the best/most interesting way to state it? Would it be better to state it in differential form, e.g., as a statement about the curvature of the field line? Is it true? Is it only true if we add some conditions? Do you need some kind of additional regularity condition?
My initial thought was that since the Lorentz transformation operates linearly on both space and the field tensor, clearly this must be true. Actually I don't think this holds up at all under examination.
As an example of a textbook treatment of this kind of thing, see Purcell, Electricity and Magnetism (3rd ed.), section 5.6, which gives a full, grotty derivation in the case of a point charge.