Using Buzzard N-body simulations, we quantify the triaxiality distribution of galaxy clusters identified with the redMaPPer optical cluster finding algorithm. We test if redMaPPer selection biases both the clusters’ shape and orientation and find that redMaPPer only biases cluster orientation towards having their major axes preferentially oriented along the line of sight. We quantify the boosting of the observed redMaPPer richness for clusters of different orientations. Modeling the richness-mass relation as log-linear we find that the log-richness amplitude ln(A) is boosted from the lowest to highest orientation bin by an amount of Δln(A) = 0.277, significant at 14-σ. In addition, the lensing profiles of halos at different orientation bins resemble a “bottleneck” shape that can be quantified with a Cauchy function. We test the correlation of triaxiality with two other leading systematics in DES Y1 cluster cosmology — miscentering and projection, and find a null correlation. Triaxiality can hence be treated an independent systematic bias to forward model. The analytic templates for the triaxiality bias on observed-richness and lensing profiles are mapped as corrections to the DES observable of richness-binned lensing profiles for redMaPPer clusters, leading to a correction of richness-binned stacked weak lensing mass of 1–5%. The mass bias confirms the DES Y1 predictions that triaxiality is a leading bias, and the richness-dependence confirms that it, along with other known systematics, do not fully resolve the tension at low richness ranges for the weak lensing mass of DES Y1 cluster cosmology with other probes. Our model should be useful for studying the impact of triaxiality bias on cosmological constraints for upcoming weak lensing surveys of galaxy clusters.