I am basing this on the current most widely accepted flat earth model, the one found on the wiki. The map is this:
The theory states that the sun acts as a spotlight, illuminating a circle of area on the flat earth while moving across it.
However, this only matches summer day-night cycles. For example, during the spring and autumn equinox, at some point, the day-night on earth would have to look like this:
And even worse, during the winter solstace, the day-night would look like this:
A simple spotlight could not make these paterns, so the spotlight theory of the sun simply does not work.