With advances in Internet of Things (IoT) technologies, billions of devices are becoming connected, which can result in the unprecedented sensing and control of the physical environments. IoT devices have diverse quality of service (QoS) requirements, including data rate, latency, reliability, and energy consumption. Meeting the diverse QoS requirements presents great challenges to existing fifth-generation (5G) cellular networks, especially in unprecedented scenarios in 5G networks, such as connected vehicle networks, where strict data packet latency may be required. The IoT devices with these scenarios have higher requirements on the packet latency in networking, which is essential to the utilization of 5G networks. In this paper, we propose a multi-tier cellular-based IoT network to address this challenge, with a particular focus on meeting application latency requirements. In the multi-tier network, access points (APs) can relay and forward packets from IoT devices or other APs, which can support higher data rates with multi-hops between IoT devices and cellular base stations. However, as multiple-hop relaying may cause additional delay, which is crucial to delay-sensitive applications, we develop new schemes to mitigate the adverse impact. Firstly, we design a traffic-prioritization scheduling scheme to classify packets with different priorities in each AP based on the age of information (AoI). Then, we design different channel-access protocols for the transmission of packets according to their priorities to ensure the QoS in networking and the effective utilization of the limited network resources. A queuing-theory-based theoretical model is proposed to analyze the packet delay for each type of packet at each tier of the multi-tier IoT networks. An optimal algorithm for the distribution of spectrum and power resources is developed to reduce the overall packet delay in a multi-tier way. The numerical results achieved in a two-tier cellular-based IoT network show that the target packet delay for delay-sensitive applications can be achieved without a large cost in terms of traffic fairness.