How to free resources and dispose injected service in ASP.NET 5/Core by the end of request?

Like the all other DI containers, it will dispose IDisposable instances for you with respecting life time of instance.

In your stuation, if instance is registered as Scoped (Instance Per Request). It will dispose this instance after request is completed.

Edit:In official documents they don't mention this.So Let's check source code to be sure:

When a scope is created, ServiceScopeFactory returns a new ServiceScope which is depended with ServiceProvider and disposable.

ServiceProvider has private List<IDisposable> _transientDisposables; which keeps disposable services when TransientCallSite is invoked in CaptureDisposable method. Also ServiceProvider has private readonly Dictionary<IService, object> _resolvedServices = new Dictionary<IService, object>(); which keeps all services for Scoped.

When liftime/scope finishes, the ServiceScope is disposed. Then it disposes ServiceProvider which disposes all _transientDisposables and then it checks _resolvedServices and disposes disposable services in the dictionary in ServiceProvider.

Edit(13.06.2017): They mention in official documents now. Service Lifetimes

When using AddScoped, it's by design that the object will have its lifetime associated with the Request.

I see no one mentioned this yet, but besides implementing IDisposable in your type, you can also use {HttpContext}.Response.RegisterForDispose(objectToDispose). Typically this is used to register an object at the start of a request (such as a controller action) to be disposed when the request ends.