Low-altitude communication networks (LACNs) serve as the critical infrastructure of the emerging low-altitude economy (LAE), supporting services such as drone delivery and infrastructure inspection. However, LACNs operate in highly dynamic three-dimensional (3D) environments characterized by high mobility and predominantly line-of-sight (LoS) propagation, creating strong coupling among key performance objectives including coverage, interference mitigation, handover management, and sensing capability. Isolated tuning of individual objectives cannot capture these cross-objective interactions, rendering conventional approaches based on experience-driven tuning and repeated field trials inefficient and costly. To address these challenges, we propose DT-MOO, a Digital Twin-based Multi-Objective Optimization framework for LACNs. By constructing a high-fidelity virtual replica that integrates realistic environmental models, electromagnetic (EM) propagation, and traffic dynamics within a unified environment, DT-MOO enables joint evaluation and systematic optimization of interdependent network parameters, scoring candidate configurations by their combined effect on multiple objectives. As the foundational validation of the framework, we report real-world experiments in a 5G-enabled LACN focusing on coverage-interference co-optimization, where DT-MOO increases the high-quality coverage rate from 14.0% to 52.9% across all evaluated altitudes compared to an operator-provisioned, experience-based baseline, while achieving a net SINR gain under stringent criteria despite local spatial trade-offs, confirming its ability to handle coupled objectives in practical LACN deployment.