When iOS 10 be released, will I be able to submit to the app store app that runs with Swift 2.2 and was built with Xcode 7.3? Or is it mandatory to move to swift 2.3 or Swift 3 and build the app with Xcode 8?
Yes you can.
Even macOS Sierra will support Xcode 7.3 (unlike Xcode 6 on El Capitan), so you can choose to build your applications in Swift 2.2 or Swift 3.
For the App Store submission, the minimum Xcode version is 6.0. (October 2016)
For more information you can check out this answer: Minimum Xcode version to upload to App store (As on February 2016)