I have a site collection that use an ASP .NET Memebership provider to authenticate user. I created an high trusted provider hosted application. But this application loses the current user authentication. Does somebody know why?
The authentication of users is very different depending on whether your Provider Hosted App is working with Office 365 or SharePoint on-premises.
In this case both SharePoint and your app should be set up to trust Azure ACS (The SharePoint part of this is handled already), so SharePoint can issue an token which is passed to the App which can use this to get a token to access SharePoint as the current user.
In this you have two options. Either low-trust which is like the Office-365 case above, but requires an Office 365 and additional configuration or high the more usual high-trust as described below.
In this case the trust in one-way SharePoint isn't going to pass any token to the app, but is going to accept what the app is telling it is the current user, so it's the responsibility of your app to authenticate the user and then pass the id of that user on to SharePoint.
This means that you should put your Membership provider into a IP-STS which both SharePoint and your app use as identity provider.