Felt, flannel, fibrous papers will get caught in your blade housing. So, it's really not worth cutting felt because it can damage your machine in the long run but yes you can cut felt and it does cut beautifully but there is a trick to it. I learned this over at the Cricut Message Board.
Use art felt. The kind of felt that is very stiff and made like heavy paper. You will need the deep housing and deep cut blade. I think you need to do a multi-cut of three if I remember right. The results are beautiful.
Also one of the former big wigs at ProvoCraft used to cut felt using this method with her Cricut. She also used the above cutting recipe.
I do want to add another thing about cutting felt. A lot of mixed media artists stiffen their felt with different type of mediums. I have read that some will run the felt through their Cricut after applying the mediums and letting it dry. Most of the mediums will stiffen felt to a paper like consistency. I do not recommend this method either. I have two Cricut's one for my personal work and one for testing purposes. I ran a normal piece of cardstock after applying a gel medium one time and almost wrecked my test Cricut. She just started acting crazy. The blade did not like that at all. Luckily, my Cricut was not hurt. Now, I cut and then apply my mediums. Don't go the medium route for felt and cutting with Cricut or the other die cut machines.
Interfacing the felt does not work either unless there is an art interfacing that I am not aware of that provides the same texture as the art felt sold in stores.
I think I covered it all.
Hope that helps.