TeamCity NUnit runner fails because of x86 - x64 - MSIL discrepancies

Based on the more detailed log you provided, the real cause of build failure appears to be:

...  
System.IO.FileNotFoundException: Could not load file or assembly 'System.Web.Http, Version=5.2.3.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35 or one of its dependencies.  
The system cannot find the file specified.  
File name: 'System.Web.Http, Version=5.2.3.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35|'  

Server stack trace:  
    at System.Reflection.RuntimeAssembly.GetExportedTypes(RuntimeAssembly assembly, ObjectHandleOnStack retTypes)  
    at System.Reflection.RuntimeAssembly.GetExportedTypes()  
    at NUnit.Core.CoreExtensions.InstallAdhocExtensions(Assembly assembly)  
    at NUnit.Core.Builders.TestAssemblyBuilder.Load(String path)  
    at NUnit.Core.Builders.TestAssemblyBuilder.Build(String assemblyName, Boolean autoSuites)  
    at NUnit.Core.Builders.TestAssemblyBuilder.Build(String assemblyName, String testName, Boolean autoSuites)  
    at NUnit.Core.TestSuiteBuilder.Build(TestPackage package)  
    at NUnit.Core.SimpleTestRunner.Load(TestPackage package)  
    at NUnit.Core.ProxyTestRunner.Load(TestPackage package)  
    at NUnit.Core.ProxyTestRunner.Load(TestPackage package)  
    at NUnit.Core.RemoteTestRunner.Load(TestPackage package)  
    at System.Runtime.Remoting.Messaging.StackBuilderSink._PrivateProcessMessage(IntPtr md, Object|[|] args, Object server, Object|[|]& outArgs)  
    at System.Runtime.Remoting.Messaging.StackBuilderSink.SyncProcessMessage(IMessage msg)  
...  

And I believe it is happening exactly because of the reason mentioned in the SO thread I have linked in the comments, i.e. you are running tests from obj folder while it should be bin. But this is still only a guess, because you are not providing your NUnit step configuration.

To address the issue you should be explicit about the paths where the assemblies with the unit tests are located. Something like this:

**\bin\**\*.Tests.dll

under "List assembly files:". That way assemblies from obj folders will not get tested, see this SO thread.