Taping felt to the door will work fine but IF you are trying to make something more permanent, and something you can pin directly into, one idea would be to make an over-the-door mounted method like the ruler organizer I made. Only instead, glue builder's foam to the top of the melamine it and then felt. OR perhaps you could use a strong double-stick tape to mount builder's foam directly onto the door after covering the face with felt. A double stick tape made for holding carpet is very strong -- likely found at Home Depot type stores.
Here's a look at the ruler thing and how I made it if that helps:
http://www.flickr.com/photos/athomesewing/2672771818/