How can I keep an app open in the background?

Solution 1:

Apps are paused when backgrounded and resume when opened. If an app does not correctly resume to the state it was left in, this is something only the developer can fix.

It's not possible to run an app in the background as if it was in the foreground. Apps can only run a limited set of tasks in the background for a short period of time per Apple guidelines, or use structured backgrounding such as Background App Refresh.

Also see iOS 7 seems to be killing apps too often.

Solution 2:

If you're on iOS 9.0 - 9.1, you can jailbreak with PanGu. You can then install a tweak called Dissident in Cydia that enables backgrounding with different behaviors per-application.