Real-world reinforcement learning (RL) offers a promising approach to training precise and dexterous robotic manipulation policies in an online manner, enabling robots to learn from their own experience while gradually reducing human labor. However, prior real-world RL methods often assume that human interventions are optimal across the entire state space, overlooking the fact that even expert operators cannot consistently provide optimal actions in all states or completely avoid mistakes. Indiscriminately mixing intervention data with robot-collected data inherits the sample inefficiency of RL, while purely imitating intervention data can ultimately degrade the final performance achievable by RL. The question of how to leverage potentially suboptimal and noisy human interventions to accelerate learning without being constrained by them thus remains open. To address this challenge, we propose SiLRI, a state-wise Lagrangian reinforcement learning algorithm for real-world robot manipulation tasks. Specifically, we formulate the online manipulation problem as a constrained RL optimization, where the constraint bound at each state is determined by the uncertainty of human interventions. We then introduce a state-wise Lagrange multiplier and solve the problem via a min-max optimization, jointly optimizing the policy and the Lagrange multiplier to reach a saddle point. Built upon a human-as-copilot teleoperation system, our algorithm is evaluated through real-world experiments on diverse manipulation tasks. Experimental results show that SiLRI effectively exploits human suboptimal interventions, reducing the time required to reach a 90% success rate by at least 50% compared with the state-of-the-art RL method HIL-SERL, and achieving a 100% success rate on long-horizon manipulation tasks where other RL methods struggle to succeed. Project website: https://silri-rl.github.io/.