If your API's data layer is sufficiently abstracted, so that you can receive consistent and predictable output for a given input, your test cases should definitely include some common output expectations for some finite list of inputs They should also test for any error cases you can drum up.
If your API can't run against a data layer that's in a predictable state (for example, if it's connected to live data, or to data that's shared among developers), you're going to end up spending a lot of time fixing your tests to correctly model the new state of the data. That makes unit testing less valuable, since you might not run them as often, and since you've never sure whether a test failure is due to a change in the data or a change in the program logic. Whether that outweighs the risk of not having unit tests or not will depend on your individual situation (how often do the tests break, how critical is the service, &ct).
As far as how to use PHPUnit to run test cases, I can't answer that specifically, but I imagine that you'd do as @basiljames noted; execute a call against a given endpoint and verify that the response received correctly matches your expectations. Not that different from any other unit test, except for the fact that they may run a little more slowly.
