High-resolution tactile sensing can provide accurate information about local contact in contact-rich robotic tasks. However, the deployment of such tasks in unstructured environments remains under-investigated. To improve the robustness of tactile robot control in unstructured environments, we propose and study a new concept: \textit{tactile saliency} for robot touch, inspired by the human touch attention mechanism from neuroscience and the visual saliency prediction problem from computer vision. In analogy to visual saliency, this concept involves identifying key information in tactile images captured by a tactile sensor. While visual saliency datasets are commonly annotated by humans, manually labelling tactile images is challenging due to their counterintuitive patterns. To address this challenge, we propose a novel approach comprised of three interrelated networks: 1) a Contact Depth Network (ConDepNet), which generates a contact depth map to localize deformation in a real tactile image that contains target and noise features; 2) a Tactile Saliency Network (TacSalNet), which predicts a tactile saliency map to describe the target areas for an input contact depth map; 3) and a Tactile Noise Generator (TacNGen), which generates noise features to train the TacSalNet. Experimental results in contact pose estimation and edge-following in the presence of distractors showcase the accurate prediction of target features from real tactile images. Overall, our tactile saliency prediction approach gives robust sim-to-real tactile control in environments with unknown distractors. Project page: https://sites.google.com/view/tactile-saliency/.