If an atheist were to say a prayer in a mocking and derisive fashion, would that have any value at all? Would it potentially incur a karmic liability? Is this a scenario where they would be better off not saying the prayer as opposed to saying it irreverently?

Your analysis is astute and accurate as to the energetics involved with such an undertaking. To make a mockery of prayer is non-divine and is also a karmic misstep as, in a sense, it is mounting a psychic attack against the divine through the negativity of the act itself, let alone the words themselves. We are assuming one is mouthing the words for a typical kind of prayer but doing so with an irreverent tone and a lack of belief by treating it with disdain under the assumption it is a silly notion to begin with and a kind of foolishness that is more deserving of mockery than serious consideration and reverence. While not the greatest of transgressions, anything one does that is negative will have a consequence and that is true here of one who mocks prayer in some way. The greatest of penalties will be what it does to the self, in closing the door on the divine through a dramatic gesture of that kind, because it holds a rather strong intention that not only is God not lofty but likely does not exist and not worthy of consideration in any respect—that is clearly saying "no" to God in a categorical way—that is cutting off your lifeline to the flow of love and light and is a dangerous direction to head because it might be your undoing given enough time with continuing neglect.