Our website content and mobile app content are the same. Will Google assess a duplicate content penalty against our web site as a result of this?
This depends on what you class as an "App" if you class a mobile version of your website i.e. m.yoursite.com then yes this will be classed as duplicate content. However, if you mean an app in the Google App Store then no this will not count as duplicate content. If you are however refering to the first, then this can be resolved easily using canonical and alternate links.
On the desktop page, add:
<link rel="alternate" media="only screen and (max-width: 640px)" href="http://m.example.com/page-1">
and on the mobile page, the required annotation should be:
<link rel="canonical" href="http://www.example.com/page-1">
rel="canonical" tag on the mobile URL pointing to the desktop page is required.
No, the content inside the mobile app is not indexed by Google. The Google App search is a search of available apps in the Google Play store and the duplicate content penalty doesn't come into play here.