Yes, you CAN run your unit tests or any other VS test type on a machine without having Visual Studio IDE installed on that machine. You need the Visual Studio Team Agent to be installed on your test machine to run your tests. This is a 15 min lightweight install and is pretty easy to configure.
Those of you familiar with remote testing from our previous versions, will already know that you can trigger a test run on the client to run tests on a remote machine with the test agent installed. Just use the test settings to point to a test agent where your tests can run remotely. With the VS Team Agent installed on machine A and the client (VS Ultimate or Pro or Test Elements) installed on machine B, you can run tests remotely on A by triggering the run from client B.
Another alternate scenario where you would want to run tests without having the Visual Studio IDE installed is during builds. If you use Team Build to do builds and run tests as part of your build process, you can setup the build agent on a machine with no VS IDE installed. You can now go ahead and run all your tests without any additional installs on the build machine in VS 2010.
What if you want to run mstest from the command line though? You can choose to run mstest.exe from either the test agent or the build agent to run tests, but note that you can run only the unit test and ordered test types.