Not all psychiatric medications require this kind of testing. It depends on the medication. The class of drugs known as selective serotonin reuptake inhibitors (SSRI), are usually given without testing blood levels for toxicity. Examples of these include Prozac and Paxil. It is possible that a manifestation of certain side effects might prompt a testing for blood levels, but for most people, this is not done.
There are other psychiatric medications that do require blood tests, for toxicity and for other reasons. One that requires fairly frequent testing is lithium because lithium toxicity is bad news and people who lose weight or become dehydrated can have elevated levels fairly quickly. There are symptoms to watch out for, including slow motion, very slurred speech, or an inability to speak in complete and logical sentences. But frequent testing should prevent any of these symptoms from occurring. Another psychiatric medication that requires blood testing is Abilfy. Unfortunately, this medication can cause the onset of diabetes, not a desirable outcome for anyone. So blood testing is performed every so often to make sure that the patient is not on the way to being diabetic.
These are just a few examples of the medications that people take for psychiatric conditions, and I am sure you can find other examples that do require testing and that do not.