The commonality here is the prefix "meta", which means self-referential. But to accurately answer the question, let's further examine these two concepts:
Metalepsis is when a figure of speech is used in a new context, or is referred to in a way that implies understanding. Consider this figure of speech: "You have to break a few eggs to make an omelet." This is a figure of speech that is widely understood to mean that sometimes you have to make a mess of things to create something new and worthwhile. So if you were to use metalepsis, you might say something like, "I've got a lot of eggs to break," which implies an understanding of the previous figure of speech. In this way, the speaker is saying that he or she intends to make something new, and is prepared to do whatever it takes to get it done. But the metalepsis does not carry if the original figure of speech isn't understood.
Metatheatre is not quite so easily defined, as there is still much debate surrounding it's broader meaning. But the metatheatrical refers to itself in some way. A prime example of metatheatre is in Hamlet, when the roving band of players come to perform for the king and end up performing a story of the murder. This is metatheatrical in two ways: first, it is a play within a play, which is in itself very metatheatrical; second, the play within a play refers to the actual story of the murder.
So, the two are related in that something that is metatheatrical only works under the same constraints as metalepsis: that is to say, it implies a greater contextual understanding.