I don’t think so… though it helps, as some of the jokes are Shakespeare “in-jokes.” My wife isn’t much of a Shakespeare buff, and she got into it too.
I think the part that saves them here is they seem to focus more on the well-known material (Hamlet, Romeo and Juliet, etc.) than on the obscure stuff. They don’t ignore the obscure things, but they keep it pretty basic for those. They explore the well-known plays in more depth.
And there’s lots of physical comedy and general wackiness to keep it moving too. I think most anyone can enjoy it, but knowing Shakespeare adds an extra dimension.