We introduce HammerDrive, a novel architecture for task-aware visual attention prediction in driving. The proposed architecture is learnable from data and can reliably infer the current focus of attention of the driver in real-time, while only requiring limited and easy-to-access telemetry data from the vehicle. We build the proposed architecture on two core concepts: 1) driving can be modeled as a collection of sub-tasks (maneuvers), and 2) each sub-task affects the way a driver allocates visual attention resources, i.e., their eye gaze fixation. HammerDrive comprises two networks: a hierarchical monitoring network of forward-inverse model pairs for sub-task recognition and an ensemble network of task-dependent convolutional neural network modules for visual attention modeling. We assess the ability of HammerDrive to infer driver visual attention on data we collected from 20 experienced drivers in a virtual reality-based driving simulator experiment. We evaluate the accuracy of our monitoring network for sub-task recognition and show that it is an effective and light-weight network for reliable real-time tracking of driving maneuvers with above 90% accuracy. Our results show that HammerDrive outperforms a comparable state-of-the-art deep learning model for visual attention prediction on numerous metrics with ~13% improvement for both Kullback-Leibler divergence and similarity, and demonstrate that task-awareness is beneficial for driver visual attention prediction.