Damage, like break the phone? No. Are you going to like your phone after doing that? No.
First I'm not even sure it's possible to completely cut off all background data, but let's say that you were successful and have eliminated all apps running in the background. What does this mean? It means that apps that would normally update little things, such as texts, emails, notifications, timelines, status are now going to have to be open in order to do these things. Which means that if you want to see your emails you will have to open the app, wait for it to update, if you're in an area with service, before you can see your latest messages. Texts will probably still come through... because I don't think they work like other apps with BG data, but you may have issues with MMS messages if the app cannot run in the background. Not that I'm a big fan of social networks, but updates to your facebook, messenger, twitter, G+ accounts will all be affected and not update until and only when the app is open. If you look at some typical data usage profiles you will see that many of these apps don't really use that much data, per day or whatever. May I assume that you are on a small and/or limited data plan? If so I might try to find another way to limit data usage before simply turning everything off.