I used to chew on my fingers, nails, and lips. I stopped chewing on my fingers and nails but I still do with my lips.
I can't remember exactly when or how I stopped eating the skin on my fingers and my nails. One thing that I think helped me is wearing mittens or gloves (obviously not out in public, unless it's actually appropriate). But since it is usually a subconscious thing and you don't even notice when you are doing it...the mittens/gloves will force you to stop. If it is possible, when out in public, try to constantly be holding something or just try to remind yourself not to bite. There's nothing particularly unhealthy about it except that your fingers might be dirty and you are letting in germs, and also it tends to look unsightly after a while on your fingers.
I haven't eaten my fingers/nails for a couple of years I think already. I haven't stopped biting my lips though, which if unfortunate because they often start bleeding, tearing, and chapping.
If you find out a way to control the lip biting, please respond =). Good luck!