We describe catalog-level simulations of Type Ia supernova (SN~Ia) light curves in the Dark Energy Survey Supernova Program (DES-SN), and in low-redshift samples from the Center for Astrophysics (CfA) and the Carnegie Supernova Project (CSP). These simulations are used to model biases from selection effects and light curve analysis, and to determine bias corrections for SN~Ia distance moduli that are used to measure cosmological parameters. To generate realistic light curves, the simulation uses a detailed SN~Ia model, incorporates information from observations (PSF, sky noise, zero point), and uses summary information (e.g., detection efficiency vs. signal to noise ratio) based on 10,000 fake SN light curves whose fluxes were overlaid on images and processed with our analysis pipelines. The quality of the simulation is illustrated by predicting distributions observed in the data. Averaging within redshift bins, we find distance modulus biases up to 0.05~mag over the redshift ranges of the low-z and DES-SN samples. For individual events, particularly those with extreme red or blue color, distance biases can reach 0.4~mag. Therefore, accurately determining bias corrections is critical for precision measurements of cosmological parameters. Files used to make these corrections are available at this https URL.