Domain specific neural network accelerators have garnered attention because of their improved energy efficiency and inference performance compared to CPUs and GPUs. Such accelerators are thus well suited for resource-constrained embedded systems. However, mapping sophisticated neural network models on these accelerators still entails significant energy and memory consumption, along with high inference time overhead. Binarized neural networks (BNNs), which utilize single-bit weights, represent an efficient way to implement and deploy neural network models on accelerators. In this paper, we present a novel optical-domain BNN accelerator, named ROBIN, which intelligently integrates heterogeneous microring resonator optical devices with complementary capabilities to efficiently implement the key functionalities in BNNs. We perform detailed fabrication-process variation analyses at the optical device level, explore efficient corrective tuning for these devices, and integrate circuit-level optimization to counter thermal variations. As a result, our proposed ROBIN architecture possesses the desirable traits of being robust, energy-efficient, low latency, and high throughput, when executing BNN models. Our analysis shows that ROBIN can outperform the best-known optical BNN accelerators and also many electronic accelerators. Specifically, our energy-efficient ROBIN design exhibits energy-per-bit values that are ~4x lower than electronic BNN accelerators and ~933x lower than a recently proposed photonic BNN accelerator, while a performance-efficient ROBIN design shows ~3x and ~25x better performance than electronic and photonic BNN accelerators, respectively.