Understanding the mechanisms governing the structure and dynamics of flexible polymers like chromosomes, especially, the signatures of motor-driven active processes is of great interest in genome biology. We study chromosomes as a coarse-grained polymer model where microscopic motor activity is captured via an additive temporally persistent noise. The active steady state is characterized by two parameters: active force, controlling the persistent-noise amplitude, and correlation time, the decay time of active noise. We find that activity drives dynamic compaction, leading to a globally collapsed entangled globule for long correlation times. Diminished topological constraints destabilize the entangled globule, and the polymer segments trapped in the globule move toward the periphery, resulting in an enriched density near the periphery. We also show that heterogeneous activity may lead to the segregation of the highly dynamic species from the less dynamic one. Our model suggests correlated motor forces as a factor (re)organizing chromosome compartments and driving transcriptionally active regions towards the chromosome periphery. This contrasts the passive adhesive or repulsive forces shaping chromosome structures. Importantly, structural ensembles are not sufficient to distinguish between the active or passive mechanisms, but the dynamics may hold key distinguishing signatures. The motor-driven polymer shows distinctive dynamic features like enhanced apparent diffusivity and exploration of all the dynamic regimes (sub-diffusion, effective diffusion, and super-diffusion) at various lag times.