What is the purpose of diodes in flip-dot displays?

If we redraw your circuit without the diodes, we have this:

^{simulate this circuit – Schematic created using CircuitLab}

Now imagine you drive Column-1 High and Row-1 Low (and Column-2 and Row-2 are Hi-Z) because you're trying to activate L1 in one direction.
You'll find that you'll also have current flowing through L2, L4 & L3 since the 3 of them in series are connected in parallel with L1.

The same circuit, drawn a little differently, but keeping all of the connections the same could look like this:

^{simulate this circuit}

and now its much easier to see how this would work.

So, the diodes are there to isolate the coils form each other to prevent this situation and allow each coil to be driven independently.