No, that wasn't it at all

The test configuration earlier today wasn't the problem. It turned out that MSBuild simply didn't know it had to pull in the System.Web.Providers assembly. Fortunately, this guy suggested a way to do it. I created a new file called AssemblyInit that looks like this:

using System.Diagnostics;
using System.Web.Providers;
using Microsoft.VisualStudio.TestTools.UnitTesting;

namespace MyApp
{
   public class AssemblyInit
   {
      [AssemblyInitialize]
      public void Initialize()
      {
         Trace.WriteLine("Initializing System.Web.Providers");
         var dummy = new DefaultMembershipProvider();
         Trace.WriteLine(string.Format("Instantiated {0}", dummy));
      }
   }
}

That does nothing more than create a hard reference to System.Web.Providers, causing MSBuild to affirmatively import it.

Now all my CI build works, the unit tests work, and I can go have a weekend.