Although this is an unsettled area of law, it might depend on the test. Many widely used psychological tests ask questions that invade the test-taker's privacy. And the jury is still out as to whether these tests have any ability to predict whether a particular employee will lie, steal, manage poorly, or cause other workplace problems. Because of these limitations, it's a good idea to talk to a lawyer before imposing a testing requirement.
For more information, see Nolo's article Testing Employees.