The evidence for this is found to a large degree in Plato`s Cratylus and Theaetetus. The main problem Heraclitus poses for Platonic epistemology is that the doctrine of flux makes the Protagorean equation of knowledge with perception untenable. If things are constantly changing then it is impossible to have true knowledge of them or derive knowledge from them. One can only perceive constantly shifting phenomena -- Heraclitean physics makes it impossible, as it were, to perceive the same river twice -- and thus knowledge of a type that is constant and timeless (like the knowledge of mathematics, or the statement `cats are quadripeds`) cannot be obtained by perception of phenomena. Given this, for us to make statements about universals or abstractions, there need to be (1) timeless objects of knowledge separated from the phenomena (the Forms) and (2) some other way of gaining knowledge (i.e. the theory of recollection set forth in Phaedo).
Note that this `middle theory of forms` and `theory of recollection`are both problematized by Plato in later dialogues, especially Parmenides, Sophist, and Philebus.