Cosmological analyses of samples of photometrically identified type Ia supernovae (SNe Ia) depend on understanding the effects of contamination from core-collapse and peculiar SN Ia events. We employ a rigorous analysis using the photometric classifier SuperNNova on state-of-the-art simulations of SN samples to determine cosmological biases due to such non-Ia contamination in the Dark Energy Survey (DES) 5-yr SN sample. Depending on the non-Ia SN models used in the SuperNNova training and testing samples, contamination ranges from 0.8 to 3.5percent, with a classification efficiency of 97.799.5percent. Using the Bayesian Estimation Applied to Multiple Species (BEAMS) framework and its extension BBC (BEAMS with Bias Correction), we produce a redshift-binned Hubble diagram marginalized over contamination and corrected for selection effects, and use it to constrain the dark energy equation-of-state, w. Assuming a flat universe with Gaussian M prior of 0.3110.010, we show that biases on w are >0.008 when using SuperNNova, with systematic uncertainties associated with contamination around 10percent of the statistical uncertainty on w for the DES-SN sample. An alternative approach of discarding contaminants using outlier rejection techniques (e.g. Chauvenets criterion) in place of SuperNNova leads to biases on w that are larger but still modest (0.0150.03). Finally, we measure biases due to contamination on w0 and wa (assuming a flat universe), and find these to be >0.009 in w0 and >0.108 in wa, 5 to 10times smaller than the statistical uncertainties for the DES-SN sample.