Project A uses DllImport to invoke code in a third party DLL. Project B uses Microsoft.VisualStudio.TestTools.UnitTesting to run some test cases on Project A.
The automated tests in project B were failing as the DLL required by project A wasn't being copied into the TestResults directory used by the tests.
Adding the DeploymentItem attribute to the test classes in Project B with the path to the DLL revolved the issue.