Outside of a research setting blood tests in psychiatry are almost entirely about ruling out physical causes rather than about arriving at a diagnosis or determining a course of treatment.
Thyroid, testosterone(for males), vitamin D levels, as already mentioned might indicate a causal factor as far as mood disorder or psychotic symptoms.
Something harder to interpret is high homocysteine levels which are indicative of "methylation deficiency". Lots of journal articles about methylation and mental health, but most seem pretty academic and not oriented towards patients or clinicians given advice on what to do. Homocysteine tests were given more last decade because homocysteine is a marker for cardiovascular disease, when it was discovered that lowering homocysteine via b vitamins or other means did not decrease cardiac risk, the few practioners that did use homocysteine tests tended to give up on them. To what extent methylation deficiencies contribute to mood disorders and/or psychosis is at best a research topic- but I paid out of pocket for a homocysteine test as I might have altered diet and supplements in hopes of decreasing psychiatric symptoms if my results were high-but my results were normal rendering the question mute.
People have reported that uncovering and treating all sorts of problems resulted in a profound improvement of their psychiatric difficulties. Candida, food allergies, gluten intolerance/Celiacs, diabetes, hypoglycemia, environmental exposure to heavy metal or other toxins, latent spirochete infections like Lyme's or syphylis, on and on. In spite of all that I think its best to get a thorough physical and regular labs, then assuming nothing contradicts doing so-treat your psychiatric disorder as psychiatric. Which means accepting one's condition likely has a complex causality including both physical and psychological causes and isn't likely to show up in any obvious way in labs, MRI's, EEGs, etc
I think though if I really wanted a physically based diagnosis and treatment I would cough up the money to see someone who uses SPECT technology in a manner similar to how Dr Amen uses it- but brain imaging and interpretation is expensive- its for complicated situations or people who can not rest til they have exhausted every diagnostic means available to get to the bottom of their diagnosis. Usually a combination of anticipated expense, exhaustion, or results that are relatively OK cause people to accept a diagnosis. Pushing things further could have good results but the more levels one has pushed it beyond an ordinary practioner the less likely a special factor will be found that explains it all and renders it utterly treatable, imo. I understand the striving though, that's why I put out my two-bits.