How do I use Moq to mock an extension method?

If you can change the extension methods code then you can code it like this to be able to test:

using System;
using Microsoft.VisualStudio.TestTools.UnitTesting;
using Moq;

public static class MyExtensions
{
    public static IMyImplementation Implementation = new MyImplementation();

    public static string MyMethod(this object obj)
    {
        return Implementation.MyMethod(obj);
    }
}

public interface IMyImplementation
{
    string MyMethod(object obj);
}

public class MyImplementation : IMyImplementation
{
    public string MyMethod(object obj)
    {
        return "Hello World!";
    }
}

So the extention methods are only a wrapper around the implementation interface.

(You could use just the implementation class without extension methods which are sort of syntactic sugar.)

And you can mock the implementation interface and set it as implementation for the extensions class.

public class MyClassUsingExtensions
{
    public string ReturnStringForObject(object obj)
    {
        return obj.MyMethod();
    }
}

[TestClass]
public class MyTests
{
    [TestMethod]
    public void MyTest()
    {
        // Given:
        //-------
        var mockMyImplementation = new Mock<IMyImplementation>();

        MyExtensions.Implementation = mockMyImplementation.Object;

        var myClassUsingExtensions = new MyClassUsingExtensions();

        // When:
        //-------
        var myObject = new Object();
        myClassUsingExtensions.ReturnStringForObject(myObject);

        //Then:
        //-------
        // This would fail because you cannot test for the extension method
        //mockMyImplementation.Verify(m => m.MyMethod());

        // This is success because you test for the mocked implementation interface
        mockMyImplementation.Verify(m => m.MyMethod(myObject));
    }
}

Extension methods are just static methods in disguise. Mocking frameworks like Moq or Rhinomocks can only create mock instances of objects, this means mocking static methods is not possible.