Can I get paid for releases of new versions of my app in the App Store? Suppose that I develop an app for the iOS platform. The app is a huge success and 80% of my target audience buys it. My customers understandably expect continuous development of the app, but my situation is difficult, because with the market saturated, I can hardly acquire new customers.

Is there a possibility to offer my existing customers a paid update of my app (e.g. buying Office 2010 does not entitle the customer to Office 2013, only to service packs, security updates, ect.) or do I need to release the update as a separate update (and is this allowed in Apple's ToS)?
The App Store doesn't offer support for paid app updates. You can release new features as in-app purchases for the current app (as @Jason said).

If, however, you want to charge for a new version of the app itself, then you need to release the new version as its own app and remove the previous version from sale. A common convention is to name the new app "MyApp 2" (with the "2" as part of the name) to distinguish it, since its actual version number will be 1 since it's a new app. Yes, it's a little confusing...

One issue to be aware of with this approach is that, because MyApp 1 and MyApp 2 are separate apps, users of MyApp 1 won't get any kind of notification that MyApp 2 is available, since it's not an "update" of that app but rather a separate thing entirely.