Do chemical garden coral formations leach toxins?
Experiments to test if toxins leach from silicates do exist; however, these experiments involve dangerous chemicals.  For that reason these tests should be left to qualified lab personnel and not described here.  Performing these experiments are expensive and would need to be performed on every combination of methodologies to make a "chemical garden".  Because so many combinations exists, quality control on such an experiment would be difficult.   

Experiment aside, a "chemical garden" in a fish tank is a bad idea.  While silica is chemically insoluble,  it can be converted biologically into orthosilicic acid.  This form of silica is soluble. This process in nature is called biological weathering.  

Chromium, lead, and perhaps some of the other compounds are carcinogenic to humans.  These compounds, even in an immobilized form, are not worth the risk and should be nowhere near a fish tank.