Hi there! I'm a frequent Diana player. She's my favorite AP mid at the moment.
Katarina is a tough match-up for Diana. She can jump to you, use her spin, do some damage and get away fairly quickly. The only retaliation she has is her shield.
She has no way to pull Katarina out of her ult except for the E, and most Diana players will blow that in the initial combo.
Morgana is also a tough lane for me when I play Diana. She can pool down the minions you're using to hide behind from her bind. Also, when Diana jumps n her, she can Black Shield herself and ult. If a Diana dives you under your own tower, she is bad and should feel bad. Pop your ult, snare, enjoy.
I agree with Scruff, Heimer is also a tough lane for Diana. And Zyra, too, anyone who can out-zone her zoning Q.