The shift towards high-bandwidth networks driven by AI workloads in data centers and HPC clusters has unintentionally aggravated network latency, adversely affecting the performance of communication-intensive HPC applications. As large-scale MPI applications often exhibit significant differences in their network latency tolerance, it is crucial to accurately determine the extent of network latency an application can withstand without significant performance degradation. Current approaches to assessing this metric often rely on specialized hardware or network simulators, which can be inflexible and time-consuming. In response, we introduce LLAMP, a novel toolchain that offers an efficient, analytical approach to evaluating HPC applications' network latency tolerance using the LogGPS model and linear programming. LLAMP equips software developers and network architects with essential insights for optimizing HPC infrastructures and strategically deploying applications to minimize latency impacts. Through our validation on a variety of MPI applications like MILC, LULESH, and LAMMPS, we demonstrate our tool's high accuracy, with relative prediction errors generally below 2%. Additionally, we include a case study of the ICON weather and climate model to illustrate LLAMP's broad applicability in evaluating collective algorithms and network topologies.