I did some tests (in .Net 3.5... later I will check at home using .Net 4).
The fact is:
Getting an object as an interface and then executing the method is faster than getting a delegate from a method then calling the delegate.
Considering the variable is already in the right type (interface or delegate) and simple invoking it makes the delegate win.
For some reason, getting a delegate over an interface method (maybe over any virtual method) is MUCH slower.
And, considering there are cases when we simple can't pre-store the delegate (like in Dispatches, for example), that may justify why interfaces are faster.
Here are the results:
To get real results, compile this in Release mode and run it outside Visual Studio.
Checking direct calls twice
  00:00:00.5834988
  00:00:00.5997071
Checking interface calls, getting the interface at every call
  00:00:05.8998212
Checking interface calls, getting the interface once
  00:00:05.3163224
Checking Action (delegate) calls, getting the action at every call
  00:00:17.1807980
Checking Action (delegate) calls, getting the Action once
  00:00:05.3163224
Checking Action (delegate) over an interface method, getting both at
  every call
  00:03:50.7326056
Checking Action (delegate) over an interface method, getting the
  interface once, the delegate at every call
  00:03:48.9141438
Checking Action (delegate) over an interface method, getting both once
  00:00:04.0036530
As you can see, the direct calls are really fast.
Storing the interface or delegate before, and then only calling it is really fast.
But having to get a delegate is slower than having to get an interface.
Having to get a delegate over an interface method (or virtual method, not sure) is really slow (compare the 5 seconds of getting an object as an interface to the almost 4 minutes of doing the same to get the action).
The code that generated those results is here:
using System;
namespace ActionVersusInterface
{
    public interface IRunnable
    {
        void Run();
    }
    public sealed class Runnable:
        IRunnable
    {
        public void Run()
        {
        }
    }
    class Program
    {
        private const int COUNT = 1700000000;
        static void Main(string[] args)
        {
            var r = new Runnable();
            Console.WriteLine("To get real results, compile this in Release mode and");
            Console.WriteLine("run it outside Visual Studio.");
            Console.WriteLine();
            Console.WriteLine("Checking direct calls twice");
            {
                DateTime begin = DateTime.Now;
                for (int i = 0; i < COUNT; i++)
                {
                    r.Run();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            {
                DateTime begin = DateTime.Now;
                for (int i = 0; i < COUNT; i++)
                {
                    r.Run();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking interface calls, getting the interface at every call");
            {
                DateTime begin = DateTime.Now;
                for (int i = 0; i < COUNT; i++)
                {
                    IRunnable interf = r;
                    interf.Run();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking interface calls, getting the interface once");
            {
                DateTime begin = DateTime.Now;
                IRunnable interf = r;
                for (int i = 0; i < COUNT; i++)
                {
                    interf.Run();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking Action (delegate) calls, getting the action at every call");
            {
                DateTime begin = DateTime.Now;
                for (int i = 0; i < COUNT; i++)
                {
                    Action a = r.Run;
                    a();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking Action (delegate) calls, getting the Action once");
            {
                DateTime begin = DateTime.Now;
                Action a = r.Run;
                for (int i = 0; i < COUNT; i++)
                {
                    a();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking Action (delegate) over an interface method, getting both at every call");
            {
                DateTime begin = DateTime.Now;
                for (int i = 0; i < COUNT; i++)
                {
                    IRunnable interf = r;
                    Action a = interf.Run;
                    a();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking Action (delegate) over an interface method, getting the interface once, the delegate at every call");
            {
                DateTime begin = DateTime.Now;
                IRunnable interf = r;
                for (int i = 0; i < COUNT; i++)
                {
                    Action a = interf.Run;
                    a();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.WriteLine();
            Console.WriteLine("Checking Action (delegate) over an interface method, getting both once");
            {
                DateTime begin = DateTime.Now;
                IRunnable interf = r;
                Action a = interf.Run;
                for (int i = 0; i < COUNT; i++)
                {
                    a();
                }
                DateTime end = DateTime.Now;
                Console.WriteLine(end - begin);
            }
            Console.ReadLine();
        }
    }
}