In this study, we consider a problem of remote safety monitoring, where a monitor pulls status updates from multiple sensors monitoring several safety-critical situations. Based on the received updates, multiple estimators determine the current safety-critical situations. Due to transmission errors and limited channel resources, the received status updates may not be fresh, resulting in the possibility of misunderstanding the current safety situation. In particular, if a dangerous situation is misinterpreted as safe, the safety risk is high. We study the joint design of transmission scheduling and estimation for multi-sensor, multi-channel remote safety monitoring, aiming to minimize the loss due to the unawareness of potential danger. We show that the joint design of transmission scheduling and estimation can be reduced to a sequential optimization of estimation and scheduling. The scheduling problem can be formulated as a Restless Multi-armed Bandit (RMAB) , for which it is difficult to establish indexability. We propose a low-complexity Maximum Gain First (MGF) policy and prove it is asymptotically optimal as the numbers of sources and channels scale up proportionally, without requiring the indexability condition. We also provide an information-theoretic interpretation of the transmission scheduling problem. Numerical results show that our estimation and scheduling policies achieves higher performance gain over periodic updating, randomized policy, and Maximum Age First (MAF) policy.