I think it is pretty darn important, but of course, it depends on the app (location services etc.). My wife is really bad about leaving apps open, and every time she complains about battery life, I look and find 15 or 20 open apps. I close them and she doesn't complain again until she leaves another dozen or more running in the background. Personally, once a day, when I'm done using the phone I close everything I don't need running and I have never had a battery issue, even with the 4S.its easy to ignore apps that are 'running' in the background but how much are they actually running? will this seriously affect battery a lot? i tend to only think of it once every few days.