I did a very small app which downloads files from a server. iPhone 4 supports background processing of the app that is the app goes in background when user switches it.
Here is my question, if my small app developed on iOS 3.1 based stuff (build to phone by changing to iOS 4) is allowed to run in background, will my app still continue download procedure ? Or should I write specially for iOS 4 for using the background processing capability.
You need to explicitly write code that allows your app to run in the background. It does not happen automatically. That means that until you change your app, it will run no different than under iPhone OS 3.1.
To learn how to enable your app to run in the background, see the excellent WWDC videos on multitasking on the Apple developer website.