Visual Multi-Object Tracking (MOT) is a crucial component of robotic perception, yet existing Tracking-By-Detection (TBD) methods often rely on 2D cues, such as bounding boxes and motion modeling, which struggle under occlusions and close-proximity interactions. Trackers relying on these 2D cues are particularly unreliable in robotic environments, where dense targets and frequent occlusions are common. While depth information has the potential to alleviate these issues, most existing MOT datasets lack depth annotations, leading to its underexploited role in the domain. To unveil the potential of depth-informed trajectory refinement, we introduce DepTR-MOT, a DETR-based detector enhanced with instance-level depth information. Specifically, we propose two key innovations: (i) foundation model-based instance-level soft depth label supervision, which refines depth prediction, and (ii) the distillation of dense depth maps to maintain global depth consistency. These strategies enable DepTR-MOT to output instance-level depth during inference, without requiring foundation models and without additional computational cost. By incorporating depth cues, our method enhances the robustness of the TBD paradigm, effectively resolving occlusion and close-proximity challenges. Experiments on both the QuadTrack and DanceTrack datasets demonstrate the effectiveness of our approach, achieving HOTA scores of 27.59 and 44.47, respectively. In particular, results on QuadTrack, a robotic platform MOT dataset, highlight the advantages of our method in handling occlusion and close-proximity challenges in robotic tracking. The source code will be made publicly available at https://github.com/warriordby/DepTR-MOT.