Accurate estimation of Room Impulse Response (RIR), which captures an environment's acoustic properties, is important for speech processing and AR/VR applications. We propose AV-RIR, a novel multi-modal multi-task learning approach to accurately estimate the RIR from a given reverberant speech signal and the visual cues of its corresponding environment. AV-RIR builds on a novel neural codec-based architecture that effectively captures environment geometry and materials properties and solves speech dereverberation as an auxiliary task by using multi-task learning. We also propose Geo-Mat features that augment material information into visual cues and CRIP that improves late reverberation components in the estimated RIR via image-to-RIR retrieval by 86%. Empirical results show that AV-RIR quantitatively outperforms previous audio-only and visual-only approaches by achieving 36% - 63% improvement across various acoustic metrics in RIR estimation. Additionally, it also achieves higher preference scores in human evaluation. As an auxiliary benefit, dereverbed speech from AV-RIR shows competitive performance with the state-of-the-art in various spoken language processing tasks and outperforms reverberation time error score in the real-world AVSpeech dataset.