This paper analyzes the impact of data freshness on remote inference systems, where a pre-trained neural network infers time-varying targets based on features observed at a sensing node. The authors use information-theoretic analysis to show that the performance degradation is monotonic if the data sequence can be approximated as a Markov chain, but not necessarily if it is far from being Markovian. They propose a new "selection-from-buffer" model for pending features and design low-complexity scheduling policies to minimize inference error. For single-source, single-channel systems, an optimal scheduling policy is provided, while for multi-source, multi-channel systems, a new scheduling policy integrating Whittle index-based source selection and duality-based feature selection-from-buffer algorithms is designed. The paper also includes experimental results demonstrating the advantages of the proposed scheduling policies.This paper analyzes the impact of data freshness on remote inference systems, where a pre-trained neural network infers time-varying targets based on features observed at a sensing node. The authors use information-theoretic analysis to show that the performance degradation is monotonic if the data sequence can be approximated as a Markov chain, but not necessarily if it is far from being Markovian. They propose a new "selection-from-buffer" model for pending features and design low-complexity scheduling policies to minimize inference error. For single-source, single-channel systems, an optimal scheduling policy is provided, while for multi-source, multi-channel systems, a new scheduling policy integrating Whittle index-based source selection and duality-based feature selection-from-buffer algorithms is designed. The paper also includes experimental results demonstrating the advantages of the proposed scheduling policies.