In this work, we present polarization profiles for 23 millisecond pulsars observed at 820 and 1500 MHz with the Green Bank Telescope as part of the NANOGrav pulsar timing array. We calibrate the data using Mueller matrix solutions calculated from observations of PSRs B1929+10 and J1022+1001. We discuss the polarization profiles, which can be used to constrain pulsar emission geometry, and present both the first published radio polarization profiles for nine pulsars and the discovery of very low-intensity average profile components ("microcomponents") in four pulsars. We obtain the Faraday rotation measures for each pulsar and use them to calculate the Galactic magnetic field parallel to the line of sight for different lines of sight through the interstellar medium. We fit for linear and sinusoidal trends in time in the dispersion measure and Galactic magnetic field and detect magnetic field variations with a period of 1 yr in some pulsars, but overall find that the variations in these parameters are more consistent with a stochastic origin.