We report results from TeV gamma-ray observations of the microquasar Cygnus X-3. The observations were made with the Very Energetic Radiation Imaging Telescope Array System (VERITAS) over a time period from 2007 June 11 to 2011 November 28. VERITAS is most sensitive to gamma rays at energies between 85 GeV and 30 TeV. The effective exposure time amounts to a total of about 44 hr, with the observations covering six distinct radio/X-ray states of the object. No significant TeV gamma-ray emission was detected in any of the states, nor with all observations combined. The lack of a positive signal, especially in the states where GeV gamma rays were detected, places constraints on TeV gamma-ray production in Cygnus X-3. We discuss the implications of the results.