It's not self predication by itself that causes trouble here. By self predication alone, the form of beauty, for example, can be beautiful (thus partake of itself) and all will be fine.
The trouble is that according to Plato's theory of forms a form must be distinct from the things which partake of it, and so it cannot partake of itself (see here). (Note that a form partaking of itself is not the same as self predication.)
So if the form of beauty is beautiful, there must be another distinct form of which the form of beauty partakes, call it B1. But then B1 is supposed to be beautiful too and so partake of a distinct B2, and so on. Thus we get infinite regress.