I'm on Bell here in Canada and I got a 500MB plan. I thought that it wouldn't be enough, and I was scared that the internet would be pulling data all the time, but it doesn't. I use to switch from my 3G connection to WiFi when I first got the phone, because I was afraid I was using too much data from the web. I stopped doing that though for a whole month, just to see how much data was actually pulled from the internet and I haven't even come close to my limit. I have apps that require updating over the internet and I have to say, it doesn't pull as much data as I thought.
I'm also not surfing the web all day, and the apps I have don't require too much data updating from the internet, so I'm starting to think 500MB is too much.
So to answer your question, the internet does not run constantly. If you're not using it, it won't do anything. Only if your app requires information from the internet, then you're internet will be running (for example, downloading a video, music or podcast on to your phone from your phone....and doing it everyday).
Hope this helps.