HDMI microcontroller / DSP tutorial
Don't implement the HDMI standard on a DSP, the standard isn't public and it's a waste of your time.

There's two ways to do this that I can think of. The "hard" way: get a receiver and a transmitter IC (ADI sells them, for example). Demod the HDMI data with a reciever, and pipe the video directly to the transmitter. Push the audio into a DSP like the ADI Blackfin, do your audio magic, then send it to the transmitter. This is not a trivial task: you need to design a PCB and you'll have to deal with high-frequency design concerns.

The "easy" way: just get a small Mini-ITX PC with an HDMI capture card and write some software do do what you want.

In both cases, you'll probably have to contend with HDCP, so you won't be able to use all media.