That seems 3-part project:
1. Connect the LEDs to a computer (like the Raspberry PI) that controls it.
2. Write a program on the Raspberry Pi that accepts commands from your server.
3. Write an iPhone app that communicates with your server to send the commands.

I wonder if it's possible to skip the server and have the Raspberry PI communicate directly with an iPhone app. Maybe with adhoc WiFi or bluetooth?

Once your technical design and hardware is ready, start programming.