Network monitoring is fundamental to effective traffic engineering (TE) in quantum networks, as it provides a timely snapshot of network status. However, quantum measurement imposes intrinsic constraints: direct measurement of entangled states collapses the quantum state, and the no-cloning theorem prohibits duplication of quantum information, rendering real-time and continuous monitoring inherently challenging. Although nondestructive techniques such as weak measurement, quantum non-demolition (QND) measurement, and protective measurement have been proposed, their roles in supporting TE have not been systematically examined within a unified framework. In this paper, we develop a unified analytical framework that integrates weak, QND, and protective measurements into TE-driven quantum network monitoring, providing the first systematic network-level analysis of these measurement paradigms in the context of quantum TE. We formulate their measurement models, characterize their information gain and disturbance trade-offs, and evaluate their applicability in terms of measurement accuracy, real time response, scalability, overhead, and robustness to noise. We observe that although weak, QND, and protective measurement techniques offer theoretically compelling approaches for mitigating measurement-induced back-action, they remain constrained by significant practical limitations. Experimental complexity, scalability challenges, resource overhead, and latency considerations collectively limit their immediate suitability for deployment in real-time quantum network monitoring frameworks supporting TE. We further distinguish between directly measurable and estimable network metrics, discuss their roles in TE, and analyze feasible measurement locations and timescales under coherence constraints. In addition, we present a formal definition of quantum traffic matrices and identify four key types of network knowledge inferred from data analysis and provide a formal definition and framework for the characterization of entanglement availability. Finally, our work lays a solid foundation and outlines a clear roadmap for achieving TE in quantum networks, while carefully accounting for the fundamental limitations imposed by quantum measurements and the no-cloning theorem.