In this paper, we analyze the impact of data freshness on remote inference systems, where a pre-trained neural network blue infers a time-varying target (e.g., the locations of vehicles and pedestrians) based on features (e.g., video frames) observed at a sensing node (e.g., a camera). One might expect that the performance of a remote inference system degrades monotonically as the feature becomes stale. Using an information-theoretic analysis, we show that this is true if the feature and target data sequence can be closely approximated as a Markov chain, whereas it is not true if the data sequence is far from being Markovian. Hence, the inference error is a function of Age of Information (AoI), where the function could be non-monotonic. To minimize the inference error in real-time, we propose a new "selection-from-buffer" model for sending the features, which is more general than the "generate-at-will" model used in earlier studies. In addition, we design low-complexity scheduling policies to improve inference performance. For single-source, single-channel systems, we provide an optimal scheduling policy. In multi-source, multi-channel systems, the scheduling problem becomes a multi-action restless multi-armed bandit problem. For this setting, we design a new scheduling policy by integrating Whittle index-based source selection and duality-based feature selection-from-buffer algorithms. This new scheduling policy is proven to be asymptotically optimal. These scheduling results hold for minimizing general AoI functions (monotonic or non-monotonic). Data-driven evaluations demonstrate the significant advantages of our proposed scheduling policies.